Vector Calculus A primer
Functions of Several Variables A single function of several variables: f: R $ R, f x (, x ),, x $ = y. Partial derivative vector, or gradient, is a vector: f = y,, y x ( x $
Multi-Valued Functions A vector-valued function of several variables: f: R $ R /, f x (, x ),, x $ = y (, y ),, y /. Can be viewed as a change of coordinates, or a mapping. We get a matrix, denoted as the Jacobian: f = y ( x ( y ( x $ y $ x ( y $ x $ https://www.math.duke.edu/education/ccp/materials/mvcalc/parasurfs/para1.html
Dot Product a, b R $, a 6 b = a 7 b 7 R. We get that a 6 b = a b cos θ, where θ is the angle between the vectors. Squared norm of vector: a ) = a 6 a. Matrix multiplication resultó dot products of row and column vectors. Alternative notation: a 6 b = a, b
Dot Product A geometric interpretation: the part of a which is parallel to a unit vector in the direction of b. And vice versa! Projected vector: a = >6?? b. The part of b orthogonal to a has no effect!
Cross Product Typically defined only for R K. a b = a N b O a O b N, b Q a O b O a Q, a Q b N a N R. Or more generally: a b = a Q a N a O b Q b N b O xr yr z
Cross Product The result vector is orthogonal to both vector Direction: Right-hand rule. Normal to the plane spanned by both vectors. Its magnitude is a b = a b sin θ. Parallel vectors ó cross product zero. The part of b parallel to a has no effect on the cross product!
Also denoted as 2-tensors. M: V V R, M u, v = c. Take two vectors into a scalar. Symmetry: M u, v = M v, u Bilinear Maps Linearity: M au + bw,v = am u, v + bm w, v. The same for v for symmetry. Can be represented by n n matrices: c=u c Mv.
Directional Derivative The change in function u in the (unit) direction d e : f u = u, d Interpretation: stretch of the function in this direction: f u Often used squared: f u )
Example: Jacobian and Change of Coordinates Suppose change of coordinates G x (, x ),, x $ = y (, y ),, y $. How does function f x (, x ),, x $ transform? f(y) = G 6 f(x) Change of length in transformed direction v = G 6 u: v ) = u c G c 6 G u. Where G c 6 G is a symmetric bilinear form. Which is also a metric. http://mathinsight.org/image/change_variable_area_transformation
Vector Fields in 3D A vector-valued function assigning a vector to each point in space: f: R K R K, f p = v. Physics: velocity fields, force fields, advection, etc. Special vector fields: Constant Rotational Gradients of scalar functions: v = g. http://vis.cs.brown.edu/results/images/laidlaw-2001-qce.011.html
Integration over a Curve Given a curve C t = x t,y t, z(t), t [t o,t ( ]. And a vector field v (x, y, z) The integration of the field on the curve is defined as: q v 6 dc r s t = q v 6 s u dx dt, dy dt, dz dt dt
Conservative Vector Fields A vector field v is conservative if there is a scalar function φ so that for every curve C t,t [t o, t ( ]: Equivalently: if v = φ. q v 6 dc r = φ t ( φ t o The integral is then path independent.
Conservative Vector Fields Physical interpretation: the vector field v is the result of a potential φ. Example: the work (potential energy) W done by gravity force G = W is only dependent of the height gained\lost. Corollary: the integral of a conservative vector field over a closed curve is zero!
Definition: v = The Curl (Rotor) Operator x xq, x z xn, x xo v. Produces a vector field from a vector field. Geometric intuition: v encodes local rotation (vorticity) that the vector field (as a force) induces locally on the point. Direction: the rotation axis nr. Integral definition: v 6 nr = lim o 1 C is an infinitesemal curve around the point A is the area it encompasses. A dv 6 dc r http://www.chabotcollege.edu/faculty/shildreth/physics/gifs/curl.gif
Irrotational Fields Fields where v = 0. Also denoted Curl-free. Conservative fields => irrotational. as for every scalar φ: φ = 0 ( It is evident from the integral definition: lim o dv 6 dc Is irrotational => Conservative fields also correct? Only (and always) for simply-connected domains! r.
Divergence Definition: 6 v = x xq, x z xn, x xo 6 v. Produces a scalar value from a vector field. Geometric intuition: 6 v encodes local change in density induced by vector field as a flux. Integral definition: 1 6 v = lim ƒ V v 6 nr ( ) S(V) is the surface of an infinitesimal volume around the point. nr is the outward local normal. http://magician.ucsd.edu/essentials/webbookse8.html
Laplacian The divergence of the gradient of a scalar field: φ = ) φ = 6 φ. Produces a scalar value from a scalar field. Geometric intuition: Measuring how much a function is diffused or similar to the average of its surrounding. Found in heat and wave equations. Used extensively in signal processing, e.g. for denoising.
Stokes Theorem A more general form of the idea of conservative fields The modern definition: q dw = q w Geometric interpretation: Integrating the differential of a field inside a domain ó integrating the field on the boundary. x
Stokes Theorem Generalizes many classical results. Integrating along a curve: φ 6 dc = φ t r ( φ t o. Special case: Fundamental theory of calculus: Q F Š x dx = F x u ( F(x o ). Kelvin-Stokes Theorem: Divergence theorem: Q t v 6 dc = x v ds. Œ and many similar more. 6 v dv = v 6 nr ds x