ENTRY MULTIVARIABLE CALCULUS Author: Oliver Knill: March 2000 -March 2004 Literature: Standard glossary of multivariable calculus course as taught at the Harvard mathematics department. +------------------------------------------------------------ | acceleration +------------------------------------------------------------ The acceleration of a parametrized curve r(t) = (x(t),y(t),z(t)) is defined as the vector r''(t). It is the rate of change of the velocity r'(t). It is significant, because Newtons law relates the acceleration r''(t) of a mass point of mass m with the force F acting on it: m r''(t) = F(r(t)) . This ordinary differential equation determines completely the motion of the particle. +------------------------------------------------------------ | advection equation +------------------------------------------------------------ The advection equation u_t= c u_x is a linear partial differential equation. Its general solution is u(t,x)=f(x+ct), where f(x)=u(0,x). The advection equation is also called transport equation. In higher dimensions, it generalizes to the gradient flow u_t = c grad(u). +------------------------------------------------------------ | Archimedes spiral +------------------------------------------------------------ The Archimedes spiral is the plane curve defined in polar coordinates as r(t) = c t, where c is a constant. In Euclidean coordinates, it is given by the parametrization r(t) = (c t cos(t), c t sin(t) ). +------------------------------------------------------------ | axis of rotation +------------------------------------------------------------ The axis of rotation of a rotation in Euclidean space is the set of fixed points of that rotation. +------------------------------------------------------------ | Antipodes +------------------------------------------------------------ Two points on the sphere of radius r are called Antipodes (=anti-podal points) if their Euclidean distance is maximal 2r. If the sphere is centered at the origin, the antipodal point to (x,y,z) is the point (-x,-y,-z). +------------------------------------------------------------ | boundary +------------------------------------------------------------ The boundary of a geometric object. Examples: The boundary of an interval I= a leq x leq b is the set with two points a,b. For example, 0 leq x leq 1 the boundary 0,1. The boundary of a region G in the plane is the union of curves which bound the region. The unit disc has as a boundary the unit circle. The entire plane has an empty boundary. The boundary of a surface S in space is the union of curves which bound the surface. For example: A semisphere has as the boundary the equator. The entire sphere has an empty boundary. The boundary of a region G in space is the union of surfaces which bound the region. For example, the unit ball has the unit sphere as a boundary. A cube has as a boundary the union of 6 faces. The boundary of a curve r(t), t in a,b consists of the two points r(a),r(b). The boundary can be defined also in higher dimensions where surfaces are also called manifolds. The dimension of the boundary is always one less then the dimension of the object itself. In cases like the half cone, the tip of the cone is not considered a part of the boundary. It is a singular point which belongs to the surface. While the boundary can be defined for far more general objects in a mathematical field called "topology", the boundaries of objects occuring in multivariable calculus are assumed to be of dimension one less than the object itself. +------------------------------------------------------------ | Burger's equation +------------------------------------------------------------ The Burger's equation u_t = u u_t is a nonlinear partial differential equation in one dimension. It is a simple model for the formation of shocks. +------------------------------------------------------------ | Cartesian coordinates +------------------------------------------------------------ Cartesian coordinates in three-dimensional space describe a point P with coordinates x, y and z. Other possible coordinate systems are cylindrical coordinates and spherical coordinates. Going from one coordinate system to an other is called a coordinate change. +------------------------------------------------------------ | Cavalieri principle +------------------------------------------------------------ Cavalieri principle tells that if two solids have equal heights and their sections at equal distances have have areas with a given ratio, then the volumes of the solids have the same ratio. +------------------------------------------------------------ | change of variables +------------------------------------------------------------ A change of variables is defined by a coordinate transformation. Examples are changes between cylindrical coordinates, spherical coordinates or Cartesian coordinates. Often one uses also rotations, allowing to use a convenient coordinate system, like for example, when one puts a coordinate system so that a surface of revolution has as the symmetry axes the z-axes. +------------------------------------------------------------ | circle +------------------------------------------------------------ A circle is a curve in the plane whose distance from a given point is constant. The fixed point is called the center of the circle. The distance is the radius of the circle. One can parametrize a circle by r(t) = (cos(t),sin(t) or given as an implicit equation g(x,y)=x^2+y^2 = 1. The circle is an example of a conic section, the intersection of a cone with a plane to which ellipses, hyperbola and parabolas belong to. +------------------------------------------------------------ | cone +------------------------------------------------------------ A cone in space is the set of points x^2+y^2=z^2 in space. Also translates, scaled and rotated versions of this set are still called a cone. For example 2x^2+3 y^2 = 7 z^2 is an elliptical cone. +------------------------------------------------------------ | conic section +------------------------------------------------------------ A conic section is the intersection of a cone with a plane. Hyperbola, ellipses and parabola lines and pairs of intersecting lines are examples of conic sections. +------------------------------------------------------------ | continuity equation +------------------------------------------------------------ The continuity equation is the partial differential equation rho_t + div(rho v)=0, where rho is the density of the fluid and v is the velocity of the fluid. The continuity equation is the consequence of the fact that the negative change of mass in a small ball is equal to the amount of mass which leaves the ball. The later is the flux of the current j=v rho through the surface and by the divergence theorem the integral of div(j). +------------------------------------------------------------ | cos theorem +------------------------------------------------------------ The cos theorem relates the length of the edges a,b,c in a triangle ABC with one of the angles alpha: a^2 = b^2+c^2-2bccos(alpha) Especially, if alpha=pi/2, it becomes the theorem of Pythagoras. +------------------------------------------------------------ | critical point +------------------------------------------------------------ A critical point of a function f(x,y) is a point (x_0,y_0), where the gradient grad f(x_0,y_0) vanishes. Critical points are also called stationary points. For functions of two variables f(x,y), critical points are typically maxima, minima or saddle points realized by f(x,y) = -x^2-y^2, f(x,y)=x^2+y^2 or f(x,y)=x^2-y^2. +------------------------------------------------------------ | chain rule +------------------------------------------------------------ The chain rule expresses the derivative of the composition of two functions in terms of the derivatives of the functions. It is (f g)'(x) = f'(g(x)) g'(x). For example, if r(t) is a curve in space and F a function in three variables, then (d/dt) f(r(t)) = grad(f) . r'(t). Example. If T and S are maps on the plane, then (T S)' = T'(S) S', where T' is the Jacobean of T and S' is the Jacobean of S. +------------------------------------------------------------ | change of variables +------------------------------------------------------------ A change of variables on a region R in Euclidean space is given by an invertible map T: R to T(R). The change of variables formula int_T(R) f(x) dx = int_R f(Tx) det(T'(x)) dx allows to evaluate integrals of a function f of several variables on a complicated region by integrating on a simple region R. In one dimensions, the change of variable formula is the formula for substitution. Example: (2D polar coordinates) T(r,phi) = (xcos(theta), y sin(theta). with det(T')=r maps the rectangle 0,s x 0,2 pi into the disc. Example of 3D spherical coordinates are T(r,theta,phi) = ( rcos(theta) sin(phi), r sin(theta) sin(phi), rcos(phi)), det(T')=r^2 sin(phi) maps the rectangular region (0,s) x (0,2pi0 x (0,pi) onto a sphere of radius s. +------------------------------------------------------------ | curl +------------------------------------------------------------ The curl of a vector field F=(P,Q,R) in space is the vector field (R_y-Q_z,P_z-R_x,Q_x-P_y). It measures the amount of circulation = vorticity of the vector field. The curl of a vector field F=(P,Q) in the plane is the scalar field (Q_x-P_y). It measures the vorticity of the vector field in the plane. +------------------------------------------------------------ | curvature +------------------------------------------------------------ The curvature of a parametrized curve r(t) = (x(t),y(t),z(t)) is defined as k(t) = |r'(t) x r''(t)|/|r'(t)|^3. Examples: The curvature of a line is zero. The curvature of a circle of radius r is 1/r. +------------------------------------------------------------ | curve +------------------------------------------------------------ A curve in space is the image of a map X: t -> r(t)=(x(t),y(t),z(t)), where x(t),y(t),z(t) are three piecewise smooth functions. For general continuous maps x(t),y(t),z(t), the length or the velocity of the curve would no more be defined. +------------------------------------------------------------ | cross product +------------------------------------------------------------ The cross product of two vectors v=(v_1,v_2,v_3) and w=(w_1,w_2,w_3) is the vector (v_2 w_3-w_2 v_3,v_3 w_1-w_3 v_1,v_1 w_2-w_2 v_1). +------------------------------------------------------------ | curve +------------------------------------------------------------ A curve in three-dimensional space is the image of a map r(t)=(x(t),y(t),z(t)), where x(t),y(t),z(t) are three continuous functions. A curve in two-dimensional space is the image of a map r(t)=(x(t),y(t)). +------------------------------------------------------------ | cylinder +------------------------------------------------------------ A cylinder is a surface in three dimensional space such that its defining equation f(x,y,z)=0 does not involve one of the variables. For example, z=2 sin(y) defines a cylinder. A cylinder usually means the surface x^2+y^2=r or a translated rotated version of this surface. +------------------------------------------------------------ | derivative +------------------------------------------------------------ The derivative of a function f(x) of one variable at a point x is the rate of change of the function at this point. Formally, it is defined as lim_dx to 0 (f(x+dx)-f(x))/dx. One writes f'(x) for the derivative of f. The derivative measures the slope of the graph of f(x) at the point. If the derivative exists for all x, the function is called differentiable. Functions like sin(x) or cos(x) are differentiable. One has for example f'(x)=cos(x) if f(x)=sin(x). An example of a function which is not differentiable everywhere is f(x)=|x|. The derivative at 0 is not defined. +------------------------------------------------------------ | cylindrical coordinates +------------------------------------------------------------ cylindrical coordinates in three dimensional space describe a point P by the coordinates r=(x^2+y^2+z^2)^1/2, phi=arctan(y/x),z, where P=(x,y,z) are the Cartesian coordinates of P. Other coordinate systems are Cartesian coordinates or spherical coordinates. +------------------------------------------------------------ | determinant +------------------------------------------------------------ The determinant of a matrix A=( a b c d ) is ad-bc. The determinant of a matrix A=( a b c d e f g h i ) is a e i+ b f g + c d h - c e g - f h a - i b d. The determinant is relevant when changing variables in integration. +------------------------------------------------------------ | directional derivative +------------------------------------------------------------ The directional derivative of f(x,y,z) in the direction v is the dot product of the gradient of f with v. It measures the rate of change of f at a point P when moving trough the point (x,y,z) with velocity v. +------------------------------------------------------------ | distance +------------------------------------------------------------ The distance of two points P=(a,b,c) and Q=(u,v,w) in three dimensional Euclidean space is the square root of (a-u)^2+(b-v)^2+(c-w)^2. The distance of two points P=(a,b) and Q=(u,v) in the plane is the square root of (a-u)^2+(b-v)^2. +------------------------------------------------------------ | distance +------------------------------------------------------------ The distance between two nonparallel lines in three dimensional Euclidean space is given by the forumla d = | (v x w) . u |/| (v x w)|, where v and w are arbitrary nonzero vectors in each line and u is an arbitrary vector connecting a point on the first line to a point of the second line. +------------------------------------------------------------ | divergence +------------------------------------------------------------ The divergence of a vector field F=(P,Q,R) is the scalar field div(F) = P_x+Q_y+R_z. The value div(F)(x,y,z) measures the amount of expansion of the vector field at the point (x,y,z). +------------------------------------------------------------ | dot product +------------------------------------------------------------ The dot product of two vectors v=(v_1,v_2,v_3) and w=(w_1,w_2,w_3) is the scalar v_1 w_1+v_2 w_2+v_3 w_3. +------------------------------------------------------------ | ellipse +------------------------------------------------------------ An ellipse is the set of points in the plane which satisfy an equation (x-a)^2/A^2+(y-b)^2/B^2=1. It is inscribed in a rectangle of length A and width B centered at (a,b). Ellipses can also be defined as the set of points in the plance whose sum of the distances to two points is constants. The two fixed points are called the foci of the ellipse. The line through the foci of a noncircular ellipse is called the focal line, the points where focal axes and a noncircular ellipse cross, are called vertices of the ellipse. The major axis of the ellipse is the line segment connecting the two vertices, the minor axis is the symmetry line of the ellipse which mirrors the two focal points or the two vertices. Ellipses are examples of conic sections, the intersection of a cone with a plane. +------------------------------------------------------------ | ellipsoid +------------------------------------------------------------ An ellipsoid is the set of points in three dimensional Euclidean space, which satisfy an equation (x-a)^2/A^2+(y-b)^2/B^2+(z-c)^2/C^2=1. It is inscribed in a box of length, width and height A,B,C centered at (a,b,c). +------------------------------------------------------------ | equation of motion +------------------------------------------------------------ The equation of motion of a fluid is the partial differential equation rho Dv/dt = - grad(p) + F, where F are external forces like gravity rho g, or magnetic force j x B and Dv/dt is the total time derivative Dv/dt = v_t+v grad(v). The term - grad(p) is the pressure force. Together with an incompressibility assumption div(v)=0, these equations of motion are called Navier Stokes equations. +------------------------------------------------------------ | flux integral +------------------------------------------------------------ The flux integral of a vector field F through a surface S=X(R) is defined as the double integral of X(F).n over R, where n=X_u x X_v is the normal vector of the surface as defined through the parameterization X(u,v). +------------------------------------------------------------ | Fubinis theorem +------------------------------------------------------------ Fubinis theorem tells that int_a^b int_c^d f(x,y) dx dy = int_c^d int_a^b f(x,y) dy dx. +------------------------------------------------------------ | gradient +------------------------------------------------------------ The gradient of a function f at a point P=(x,y,z) is the vector (f_x(x,y,z),f_y(x,y,z),f_z(x,y,z)) where f_x denotes the partial derivative of f with respect to x. +------------------------------------------------------------ | Hamilton equations +------------------------------------------------------------ The Hamilton equations to a function f(x,y) is the system of ordinary differential equations x'(t) = f_y(x,y), y'(t) = - f_x(x,y). which is called Hamilton system. Solution curves of this system are located on level curves f(x,y)=c because by the chain rule one has d/dt f(x(t),y(t)) = f_x x' + f_y y' = f_x f_y - f_x f_y = 0. The preservation of f is in physics called energy conservation. +------------------------------------------------------------ | heat equation +------------------------------------------------------------ The heat equation is the Partial differential equation u_t = mu Delta(u), where mu is a constant, and Delta u is the Laplacian of u. The heat equation is also called the diffusion equation. +------------------------------------------------------------ | Hessian +------------------------------------------------------------ The Hessian is the determinant of the Hessian matrix. +------------------------------------------------------------ | Hessian matrix +------------------------------------------------------------ The Hessian matrix of a function f(x,y,z) at a point (u,v,w) is the 3x3 matrix f''(u,v,w)=H(u,v,w) = ( f_xx f_xy f_xz f_yx f_yy f_yz f_zx f_zy f_zz ). The Hessian matrix of a function f(x,y) at a point (u,v) is the 2x2 matrix f'' = H(u,v,w) = ( f_xx f_xy f_yx f_yy ). The Hessian matrix is useful to classify critical points of f(x,y) using the second derivative test. +------------------------------------------------------------ | hyperbola +------------------------------------------------------------ A hyperbola is a plane curve which can be defined as the level curve g(x,y)=x^2/a^2+y^2/b^2=1 or given as a parametrized curve r(t)=(a cosh(t),b sinh(t)). A hyperbola can geometrically also be defined as the set of points whose distances from two fixed points in the plane is constant. The two fixed points are called the focal points of the hyperbola. The line through the focal points of a hyperbola is called the focal axis. The points, where the focal axis and the hyperbola cross are called vertices. A hyperbola is an example of a conic section, the intersection of a cone with a plane. +------------------------------------------------------------ | hyperboloid +------------------------------------------------------------ A hyperboloid is the set of points in three dimensional Euclidean space, which satisfy an equation (x-u)^2/a^2-(y-v)^2/b^2-(z-w)^2/c^2= I, where I=1 or I=-1. For a=b=c=1, the hyperboloid is obtained by rotating a hyperbola x^2-y^2=1 around the x-axes. It is two-sided for I=-1 (the intersection of the plane z=c with the hyperboloid is then empty) and one-sided for I=1. +------------------------------------------------------------ | incompressible +------------------------------------------------------------ A vector field F is called incompressible if its divergence is zero div(F)=0. The notation has its origins from fluid dynamics, where velocity fields F of fluids, gases or plasma often are assumed to be incompressible. If a vector field is incompressible and is a velocity field, then the corresponding flow preserves the volume. +------------------------------------------------------------ | continuity equation +------------------------------------------------------------ The continuity equation rho_t + div(i) = 0 links density rho and velocity field i. It is an infinitesimal description which is equivalent to the preservation of mass by the theorem of Gauss. The change of mass M(t) int int int_R rho dV inside a region R in space is the minus the flux of mass through the boundary S of R. +------------------------------------------------------------ | interval +------------------------------------------------------------ An interval is a subset of the real line defined by two points a,b. One can write I = a leq x leq b for a closed interval, I = a < x < b for an open interval and I = a leq x < b , I = a < x leq b for half open intervals. If a=-infinity and b=infinity, then the interval is the entire real line. If a=0,b-infinity, then I = (a,b) is the set of positive real numbers. Intervals can be characterized as the connected sets in the real line. +------------------------------------------------------------ | integral +------------------------------------------------------------ An integral of f(x) over an interval I on the line is the limit (1/n) sum_i=1^n f(i/n) for n to infinity over the integers and the sum is taken over all i such that i/n is in I. An integral of f(x,y) over a region R in the plane is the limit (1/n^2) sum_(i/n,j/n) in R f(i/n,j/n) for n to infinity. Such an integral is also called double integral. Often, double integrals can be evaluated by iterating two one-dimensional integrals. An integral of f(x,y,z) over a domain R in space is the limit (1/n^3) sum_(i/n,j/n,k/n) in R f(i/n,j/n,k/n) for n to infinity. Such an integral is also called a triple integral. Often, triple integrals can be evaluated by iterating three one-dimensional integrals. +------------------------------------------------------------ | intercept +------------------------------------------------------------ An intercept is the intersection of a surface with a coordinate axes. Like traces, intercepts are useful for drawing surfaces by hand. For example, the two sheeted hyperboloid x^2+y^2-z^2=-1 has the intercepts x^2-z^2=-1 and y^2-z^2=-1 (hyperbola) and an empty intercept with the z axes. +------------------------------------------------------------ | jerk +------------------------------------------------------------ The jerk of a parametrized curve r(t)=(x(t),y(t),z(t)) is defined as r'''(t). It is the rate of change of the acceleration. By Newtons law, the jerk measures the rate of change of the force acting on the body. +------------------------------------------------------------ | Lagrange multiplier +------------------------------------------------------------ A Lagrange multiplier is an additional variable introduced for solving extremal problems under constraints. To extremize f(x,y,z) on a surface g(x,y,z)=0 then an extremum satisfies the equations f'= L g',g=0, where L is the Lagrange multiplier. These are four equations for four unknowns x,y,z,l. Additionally, one has to check for solutions of g'(x,y,z)=0. Example. If we want to extremize F(x,y,z)= -x log(x)-y log(y)-z log(z) under the constraint G(x,y,z)= x+y+z=1, we solve the equations -1-log(x)=L 1 -1-log(y)=L 1 -1-log(z)=L 1 x+y+z = 1, the solution of which is x=y=z=1/3. +------------------------------------------------------------ | Lagrange method +------------------------------------------------------------ The Lagrange method to solve extremal problems under constraints: 1) in order that a function f of several variables is extremal on a constraint set g=c, we either have grad g=0 or the point is a solution to the Lagrange equations grad f = L grad g, g=c. 2) in order to extremize a function f of several variables under the contraint set g=c,h=d, we have to solve the Lagrange equations grad f = L grad g + mu grad h, g=c,h=d or solve grad g = grad h = 0. +------------------------------------------------------------ | Laplacian +------------------------------------------------------------ The Laplacian of a function f(x,y,z) is defined as Delta(f)=f_xx+f_yy+f_zz. One can write it as Delta= div grad(f). Functions for which the Laplacian vanish are called harmonic. Laplacian appear often in PDE's Examples: the Laplace equation Delta(f)=0, the Poisson equation Delta(f)=rho, the Heat equation f_t=mu Delta(f) or the wave equation f_tt = c^2 Delta f. % The length of a curve r(t)=(x(t),y(t),z(t)) from t=a to t=b is the integral of the speed |r'(t)| over the interval a,b. Example. the length of the curve r(t)=(cos(t),sin(t)) from t=0 to t=pi is pi because the speed |r'(t)| is 1. +------------------------------------------------------------ | length +------------------------------------------------------------ The length of a vector v=(a,b,c) is the square root of v . v=a^2+b^2+c^2. An other word for length is norm. If a vector has length 1, it is called normalized or a unit vector. +------------------------------------------------------------ | level curve +------------------------------------------------------------ A level curve of a function f(x,y) of two variables is the set of points which satisfy the equation f(x,y)=c. For example, if f(x,y)=x^2-y^2, then its level curves are hyperbola. Level curves are orthogonal to the gradient vector field grad(f). +------------------------------------------------------------ | level surface +------------------------------------------------------------ A level surface of a scalar function f(x,y,z) is the set of points which satisfy f(x,y,z)=c. For example, if f(x,y,z)=x^2+y^2+3 z^2, then its level surfaces are ellipsoids. Level surfaces are orthogonal to the gradient field grad(f). +------------------------------------------------------------ | linear approximation +------------------------------------------------------------ The linear approximation of a function f(x,y,z) at a point (u,v,w) is the linear function L(x,y,z) = f(u,v,w) + grad f(u,v,w) . (x-u,y-v,z-w). +------------------------------------------------------------ | line +------------------------------------------------------------ A line in three-dimensional space is a curve in space given by r(t) = P + t v, where P is a point in space and v is a vector in space. The representation r(t)=P+tv is called a parameterization of the line. Algebraically, a line can also be given as the intersection of two planes: a x + b y + c z=d, u x + v y + w z = q. The corresponding vector v in the line is the cross product of (a,b,c) and (u,v,w). A point P=(x,y,z) on the line can be obtained by fixing one of the coordinates, say z=0 and solving the system a x + b y=d, u x + v y = q for the unknowns x and y. +------------------------------------------------------------ | line integral +------------------------------------------------------------ The line integral of a vector field F(x,y) along a curve C: r(t)=(x(t),y(t)), t in a,b in the plane is defined as int_C F . ds = int_a^b F(r(t)) . r'(t) dt , where r'(t)= (x'(t),y'(t)) is the velocity. The definition is similar in three dimensions where F(x,y,z) is a vector field and C: r(t)=(x(t),y(t),z(t)), t in a,b is a curve in space. +------------------------------------------------------------ | Maxwell equations +------------------------------------------------------------ The Maxwell equations are a set of partial differential equations which determine the electric field E and magnetic field B, when the charge density rho and the current density j are given. There are 4 equations: div(B) = 0 no magnetic monopoles curl(E) = -B_t/c Faradays law, change of magnetic flux produces voltage curl(B) = E_t/c+ (4pi/c) j Ampere's law, current or E change produce magnetism div(E) = 4 pi rho Gauss law, electric charges produce an electric field +------------------------------------------------------------ | nabla +------------------------------------------------------------ nabla is a mathematical symbol used when writing the gradient grad f of a function f(x,y,z). Nabla looks like an upside down Delta. Etymologically, the name has the meaning of an Egyption harp. +------------------------------------------------------------ | nabla calculus +------------------------------------------------------------ The nabla calculus introduces the vector grad=(partial_x,partial_y,partial_z). It satisfies grad(f) = grad(f), grad x F = curl(F), grad . F = div(F). Using basic vector operation rules and differentiation rules like grad (f g) = (grad f) g + f (grad g) one can verify identities: like for example div ( curl) F = 0, curl ( grad) f = 0, curl ( curl F) = grad (div F) - Delta (F), div(E x F) = F . curl(E) - E . curl(F). +------------------------------------------------------------ | nonparallel +------------------------------------------------------------ Two vectors v and w are called nonparallel if they are not parallel. Two vectors in space are parallel if and only if their cross product v x w is nonzero. +------------------------------------------------------------ | normal vector +------------------------------------------------------------ A normal vector to a parametrized surface X(u,v)=(x(u,v),y(u,v),z(u,v)) at a point P=(x,y,z) is the vector X_u x X_v. It is orthogonal to the tangent plane spanned by the two tangent vectors X_u and X_v. +------------------------------------------------------------ | normalized +------------------------------------------------------------ A vector is called normalized if its length is equal to 1. For example, the vector (3/5,4/5) is normalized. The vector (2,1) is not normalized. +------------------------------------------------------------ | octant +------------------------------------------------------------ An octant is one of the 8 regions when dividing three dimensional space with coordinate planes. It is the analogue of quadrant in two dimensions. +------------------------------------------------------------ | open set +------------------------------------------------------------ An open set R in the plane or in space is a set for which every point P is contained in a small disc U which is still contained in R. The disc x^2+y^2<1 is an example of an open set. The set x^2+y^2 leq 1 is not open because the point (1,0) for example has no neighborhood disc contained in R. +------------------------------------------------------------ | open +------------------------------------------------------------ A set is called open, if it is an open set. It means that every point in the set is contained in a neighborhood which still is in the set. The complement of open sets are called closed. +------------------------------------------------------------ | ordinary differential equation +------------------------------------------------------------ An ordinary differential equation (ODE) is an equation for a function or curve f(t) which relates derivatives f,f',f''.... of f. An example is f'=c f which has the solution f(t) = C e^(ct), where C is a constant. Only derivatives with respect to one variable may appear in an ODE. In most cases, the variable t is associated with time. Examples: f'= c f population model c>0. f'= - c f radioactive decay c>0 f'= c f (1-f) logistic equation f''=-c f harmonic oscillator f''= F(f) general form of Newton equations By increasing the dimension of the phase space, every ordinary differential equation can be written as a first order autonomous system x'=F(x). For example, f''=-f can be written with the vector x=(x_1,x_2)=(f,f') as (x_1',x_2')=(f',f'')=(f',f)=(x_2',-x_2'). There is a 2 x 2 matrix such that x'=A x. +------------------------------------------------------------ | orthogonal +------------------------------------------------------------ Two vectors v and w are called orthogonal if v . w=0. An other word for orthogonal is perpendicular. The zero vector 0 is orthogonal to any other vector. +------------------------------------------------------------ | parabola +------------------------------------------------------------ A parabola is a plane curve. It can be defined as the set of points which have the same distance to a line and a point. The line is called the directrix, the point is called the focus of the parabola. One can parametrize a parabola as r(t)=(t,t^2). It is also possible to give a parabola as a level curve g(x,y)=y-x^2=0 of a function of two variables. A parabola is an example of a conic section, to which also circles, ellipses and hyperbola belong. +------------------------------------------------------------ | parallelogram +------------------------------------------------------------ A parallelogram E can be defined as the image of the unit square under a map T(s,t) = s v + t w, where u and v are vectors in the plane. One says, E is spanned by the vectors v and w. The area of a parallelogram is |v x w|. +------------------------------------------------------------ | parallelepiped +------------------------------------------------------------ A parallelepiped E can be defined as the image of the unit cube under a linear map T(r,s,t) = r u + s v + t v, where u,v,w are vectors in space. One says, E is spanned by the vectors u,v and w. The volume of a parallelepiped is |u . (v x w)|. +------------------------------------------------------------ | perpendicular +------------------------------------------------------------ Two vectors v and w are called perpendicular if v . w=0. An other word for perpendicular is orthogonal. The zero vector v=0 is perpendicular to any other vector. +------------------------------------------------------------ | quadratic approximation +------------------------------------------------------------ The quadratic approximation of a function f(x,y,z) at a point (u,v,w) is the quadratic function Q(x,y,z) = L(x,y,z) + H(u,v,w) (x-u,y-v,z-w) . (x-u,y-v,z-w)/2, where H(u,v,w) is the Hessian matrix of f at (u,v,w) and where L(x,y,z) is the linear approximation of f(x,y,z) at (u,v,w). For example, the function f(x,y) = 3+sin(x+y)+cos(x + 2 y) has the linear approximation L(x,y) = 4+x+y and the quadratic approximation Q(x,y) = 4+x+y + (x+2y)^2/2. +------------------------------------------------------------ | quadrant +------------------------------------------------------------ A quadrant is one of the 4 regions when dividing the two dimensional space using coordinate axes. It is the analogue of octant in three dimensions. For example, the set x>0,y>0 is the open upper right quadrant. The set x geq 0, y geq 0 is the closed upper right quadrant. +------------------------------------------------------------ | parallel +------------------------------------------------------------ Two vectors v and w are called parallel if there exists a real number L such that v = L w. Two vectors in space are parallel if and only if their cross product v x w is zero. +------------------------------------------------------------ | parametrized surface +------------------------------------------------------------ A parametrized surface is defined by a map X(u,v) = (x(u,v),y(u,v),z(u,v)) from a region R in the uv-plane to xyz-space. Examples Sphere: X(u,v) = (rcos(u) sin(v), r sin(u) sin(v), rcos(v)) R = 0,2 pi) x 0,pi, u and v are called Euler angles. Plane X(u,v) = P + u U + v V, where P is a point, U,V are vectors and R is the entire plane. Surface of revolution is parametrized by X(u,v) = (f(v) cos(u), f(v) sin(u), v) where u is an angle measuring the rotation round the z axes and f(v) is a nonnegative function giving the distance to the z-axes at the height v. A graph of a function f(x,y) is parametrized by X(u,v) = (u,v,f(u,v)). A torus is parametrized by X(u,v) = (a+bcos(v))cos(u), (a+bcos(v)) sin(u),sin(v)) on R=0,2 pi) x 0,2 pi). +------------------------------------------------------------ | parametrized curve +------------------------------------------------------------ A parametrized curve in space is defined by a map r(t) = (x(t),y(t),z(t)) from an interval I to space. Examples are Circle in the xy-plane r(t) = (cos(t), sin(t), 0) with t in 0,2pi. Helix r(t) = (cos(t), sin(t), t) with t in a,b. Line r(t) = P + t V, where V is a vector and P is a point and -infinity0 and and f_xx(x,y)<0 then (x,y) is a local maximum. If det(f''(x,y))<0 and f_xx(x,y)>0 then (x,y) is a local minimum. +------------------------------------------------------------ | Space +------------------------------------------------------------ Space is usually used as an abbreviation for three dimensional Euclidean space. In a wider sense, it can mean linear space a vector space in which on can add and scale. +------------------------------------------------------------ | speed +------------------------------------------------------------ The speed of a curve r(t)=(x(t),y(t),z(t)) at time t is the length of the velocity vector r'(t)=(x'(t),y'(t),z'(t)). +------------------------------------------------------------ | sphere +------------------------------------------------------------ A sphere is the set of points in space, which have a given distance r from a point P=(a,b,c). It is the set (x-a)^2+(y-b)^2+(z-c)^2=r^2. For a=b=c=0,r=1 one obtains the unit sphere: x^2+y^2+z^2=1. Spheres can be define in any dimenesions. A sphere in two dimensions is a circle. A sphere in 1 dimension is the union of two points. The unit sphere in 4 dimensions is the set of points (x,y,z,w) in R^4 which satisfy x^2+y^2+z^2+w^2=1 Spheres can be defined in any space equiped with a distance like d((x,y),(u,v))=|x-u|+|y-v| in the plane. +------------------------------------------------------------ | superformula +------------------------------------------------------------ The superformula describes a class of curves with a few parameters m,n_1,n_2,n_3,a,b. It is the polar graph r(t) = (|cos( m t/4 )|^n_1/a + |sin( m t/4 )|^n_2/b)^-1/n_3 . It had been proposed by the Belgian Biologist Johan Gielis in 1997. +------------------------------------------------------------ | superposition +------------------------------------------------------------ The principle of superposition tells that the sum of two solutions of a linear partial differential equation (PDE) is again a solution of the PDE. For example, f(x,y) = sin(x-t) and g(x,y) = e^x-t are both solutions to the transport equation f_t(t,x) + f_x(t,x) = 0. Therefore also the sum sin(x-t)+e^x-t is a solution. For nonlinear partial differential equations the superposition principle is no more true which is one of the reasons for the difficulty with dealing with nonlinear systems. +------------------------------------------------------------ | surface +------------------------------------------------------------ A surface can either be described as a parametrized surface or implicitely as a level surface g(x,y,z) = 0. In the first case, the surface is given as the image of a map X:(u,v) mapsto (x(u,v),y(u,v),z(u,v)) where u,v ranges over a parameter domain R in the plane. In the second case, the surface is determined by a function of three variables. Sometimes, one can describe a surface in both ways like in the following examples: Sphere: X(u,v) = (rcos(u) sin(v),r sin(u) sin(v),rcos(v)), g(x,y,z) = x^2+y^2+z^2=r^2 Graphs: X(u,v) = (u,v,f(u,v)), g(x,y,z) = z-f(x,y) = 0 Planes: X(u,v) = P + u U + v V, g(x,y,z) = a x + b y + c z = d, (a,b,c)= U x V. Surface of revolution: X(u,v) = (f(v)cos(u), f(v) sin(u), v), g(x,y,z) = f( (x^2+y^2)^(1/2) ) - z = 0 +------------------------------------------------------------ | surface of revolution +------------------------------------------------------------ A surface of revolution is a surface which is obtained by rotating a curve around a fixed line. If that line is the z-axes, the surface can be given in cylindrical coordinates as r = f(z). A parametrization is X(t,z) = (f(z)cos(t), f(z) sin(t), z). +------------------------------------------------------------ | surface area +------------------------------------------------------------ The surface area of surface S=X(R) is defined as the integral of int int_R |X_u x X_v(u,v)| dudv. For example, for X(u,v) = ( r cos(u) sin(v), r sin(u) sin(v), r cos(v)) on R = 0 leq u leq 2 pi, 0 leq v leq pi , where S=X(R) is the sphere of radius r, one has X_u x X_v = r sin(v) X and |X_u x X_v| = sin(v) r^2. The surface area is int_0^2pi int_0^pi r^2 sin(v) du dv = 4 pi r^2. +------------------------------------------------------------ | surface integral +------------------------------------------------------------ A surface integral of a function f(x,y,z) over a surface S=X(R) is defined as the integral of f(X(u,v)) |X_u x X_v(u,v)| over R. In the special case when f(x,y,z)=1, the surface integral is the surface area of the surface S. +------------------------------------------------------------ | tangent plane +------------------------------------------------------------ The tangent plane to an implicitely defined surface g(x,y,z)=c at the point (x_0,y_0,z_0) is the plane a x + b y + c z = d, where (a,b,c) = grad f(x_0,y_0,z_0) is the gradient of g at (x_0,y_0,z_0) and d = a x_0 + b y_0 + c z_0. +------------------------------------------------------------ | tangent line +------------------------------------------------------------ The tangent line to an implicitely defined curve g(x,y)=c at the point (x_0,y_0) is the line a x + b y = d, where (a,b) is the gradient of g(x,y) at the point (x_0,y_0) and d = a x_0 + b y_0. +------------------------------------------------------------ | theorem of Clairot +------------------------------------------------------------ The theorem of Clairot assures that one can interchange the order of differentiation when taking partial derivatives. More precicely, if f(x,y) is a function of two variables for which both f_xy = f_yx are continuous, then f_xy = f_yx. +------------------------------------------------------------ | theorem of Gauss +------------------------------------------------------------ The theorem of Gauss states that the flux of a vector field F through the boundary S of a solid R in three-dimensional space is the integral of the divergence div(F) of F over the region R: int int int_R div(F) dV = int int_S F . dS . +------------------------------------------------------------ | theorem of Green +------------------------------------------------------------ The theorem of Green states that the integral of the curl(F)=Q_x-P_y of a vector field F=(P,Q) over a region R in the plane is the same as the line integral of F along the boundary C of R. int int_R curl(F) dA= int_C F ds . The boundary C is traced in such a way that the region is to the left. The boundary has to be piecewise smooth. The theorem of Green can be derived from the theorem of Stokes. +------------------------------------------------------------ | Green's theorem +------------------------------------------------------------ Green's theorem see theorem of Green. +------------------------------------------------------------ | Green's theorem +------------------------------------------------------------ The determinant of the Jacobean matrix is often called Jacobean or Jacobean determinant. +------------------------------------------------------------ | Jacobean matrix +------------------------------------------------------------ Jacobean matrix If T(u,v) = (f(u,v),g(u,v)) is a transformation from a region R to a region S in the plane, the Jacobean matrix dT is defined as ( f_u(u,v) f_v(u,v) g_u(u,v) g_v(u,v) ). It is the linearization of T near (u,v). Its determinant called the Jacobean determiant measures the area change of a small area element dA=dudv when maped by T. For example, if T(r,theta) = (r cos(theta), r sin(theta))=(x,y) is the coordinate transformation which maps R = r geq 0, theta in 0,2pi) to the plane, then dT is the matrix ( cos(theta) sin(theta) -r sin(theta) r cos(theta) ) which has determinant r. +------------------------------------------------------------ | theorem of Stokes +------------------------------------------------------------ The theorem of Stokes states that the flux of a vector field F in space through a surface S is equal to the line integral of F along the boundary C of S: int int_S curl(F) . dS = int_C F ds . +------------------------------------------------------------ | three dimensional space +------------------------------------------------------------ The three dimensional space consists of all points (x,y,z) where x,y,z ranges over the set of real numbers. To distinguish it from other three-dimensional spaces, one calls it also Euclidean space. +------------------------------------------------------------ | torus +------------------------------------------------------------ A torus is a surface in space defined as the set of points which have a fixed distance from a circle. It can be parametrized by X(u,v) = (a+bcos(v))cos(u), (a+bcos(v)) sin(u),sin(v)) on R=0,2 pi) x 0,2 pi), where a,b are positive constants. +------------------------------------------------------------ | trace +------------------------------------------------------------ The trace of a surface in three dimensional space is the intersection of the surface with one of the coordinate planes x=0 or y=0 or z=0. Traces help to draw a surface when given the task to do so by hand. Other marking points are intercepts, the intersection of the surface with the coordinate axes. +------------------------------------------------------------ | triangle +------------------------------------------------------------ A triangle in the plane or in space is defined by three points P,Q,R. If v=PQ,w=PR, then |v x w|/2 is the area of the triangle. +------------------------------------------------------------ | triple product +------------------------------------------------------------ The triple product between three vectors u,v,w in space is defined as the scalar u . (v x w). The absolute value |u . (v x w)| is the volume of the paralelepiped spanned by u,v and w. +------------------------------------------------------------ | triple dot product +------------------------------------------------------------ triple dot product (see triple product). +------------------------------------------------------------ | unit sphere +------------------------------------------------------------ The unit sphere is the sphere x^2+y^2+z^2=1. It is an example of a two-dimensional surface in three dimensional space. +------------------------------------------------------------ | unit tangent vector +------------------------------------------------------------ The unit tangent vector to a parametrized curve r(t)=(x(t),y(t),z(t)) is the normalized velocity vector T(t)=r'(t)/|r'(t)|. Together with the normal vector N(t) = T'(t)/|T'(t)| and the binormal vector B(t)=T(t) x N(t), it forms a triple of mutually orthogonal vectors. +------------------------------------------------------------ | vector +------------------------------------------------------------ A vector in the plane is defined by two points P,Q. It is the line segment v pointing from P to Q. If P=(a,b) and Q=(c,d) then the coordinates of the vector are v=(c-a,d-b). Points P in the plane can be identified by vectors pointing from 0 to P. A vector in space is defined by two points P,Q in space. If P=(a,b,c) and Q=(d,e,f), then the coordinates of the vector are v=(d-a,e-b,f-c). Points P in space can be identified by vectors pointing from 0 to P. Two vectors which can be translated into each other are considered equal. Remarks. One could define vectors more precisely as affine vectors and introduce an equivalence relation among them: two vectors are equivalent if they can be translated into each other. The equivalence classes are the vectors one deals with in calculus. Since the concept of equivalence relation would unnessesarily confuse students, the more fuzzy definition above is prefered. One should avoid definitions like "Vectors are objects which have length and direction" given in some Encyclopedias. The zero vector (0,0,0) is an example of an object which has length but no direction. It nevertheless is a vector. +------------------------------------------------------------ | vector field +------------------------------------------------------------ A vector field in the plane is a map F(x,y)=(P(x,y),Q(x,y)) which assigns to each point (x,y) in the plane a vector F(x,y). An example of a vector field in the plane is F(x,y) = (-y,x). An other example is the gradient field F(x,y) = grad f(x,y) where f(x,y) is a function. A vector field in space is a map F(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z)) which assigns to each point (x,y,z) in space a vector F(x,y,z). An example is the vector field F(x,y,z) = (x^2,y z,x-y). An other example is the gradient field F(x,y,z) = grad f(x,y,z) of a function f(x,y,z). +------------------------------------------------------------ | velocity +------------------------------------------------------------ The velocity of a parametrized curve r(t)=(x(t),y(t),z(t)) at time t is the vector r'(t) = (x'(t),y'(t),z'(t)). It is tangent to the curve at the point r(t). +------------------------------------------------------------ | volume +------------------------------------------------------------ The volume of a body G is defined as the integral of the constant function f(x,y,z)=1 over the body G. +------------------------------------------------------------ | wave equation +------------------------------------------------------------ The wave equation is the partial differential equation u_tt=c^2 Delta(u), where Delta(u) is the Laplacian of u. Light in vacuum satisfies the wave equation. This can be derived from the Maxwell equations: the identity Delta(B) = grad( div(B)- curl( curl(B)) gives together with div(B)=0 and curl(B)=E_t/c the relation Delta(B)=-d/dt curl(E)/c which leads with the Maxwell equation B_t=-c curl(E) to the wave equation Delta B = B_tt/c^2. The equation E_tt= c^2 Delta E is derived in the same way. +------------------------------------------------------------ | zero vector +------------------------------------------------------------ The zero vector is the vector for which all components are zero. In the plane it is v=(0,0), in space it is v=(0,0,0). The zero vector is a vector. It has length 0 and no direction. Definitions like "a vector is a quantity which has both length and direction" are misleading. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 119 entries in this file. COUNT: 119