Vector Calculus A primer
Vector CalculusA primer
Functions of Several Variables
โข A single function of several variables:
๐: ๐ ๐ โ ๐ , ๐ ๐ฅ1, ๐ฅ2, โฏ , ๐ฅ๐ = ๐ฆ.
โข Partial derivative vector, or gradient, is a vector:
๐ป๐ =๐๐ฆ
๐๐ฅ1, โฏ ,
๐๐ฆ
๐๐ฅ๐
Multi-Valued Functions
โข A vector-valued function of several variables:
๐: ๐ ๐ โ ๐ ๐, ๐ ๐ฅ1, ๐ฅ2, โฏ , ๐ฅ๐ = ๐ฆ1, ๐ฆ2, โฏ , ๐ฆ๐ .
โข Can be viewed as a change of coordinates, or a mapping.
โข We get a matrix, denoted as the Jacobian:
๐ป๐ =
๐๐ฆ1
๐๐ฅ1โฎ
๐๐ฆ1
๐๐ฅ๐
โฏ โฏ๐๐ฆ๐
๐๐ฅ1โฎ
๐๐ฆ๐
๐๐ฅ๐
https://www.math.duke.edu/education/ccp/materials/mvcalc/parasurfs/para1.html
Dot Product
โข ๐, ๐ โ ๐ ๐, ๐ โ ๐ = ๐๐ โ ๐๐ โ ๐ .
โข We get that ๐ โ ๐ = ๐ ๐ cos ๐, where ๐ is the angle between the vectors.
โข Squared norm of vector: ๐ 2 = ๐ โ ๐.
โข Matrix multiplication result dot products of row and column vectors.
Dot Product
โข A geometric interpretation: the part of ๐ which is parallel to a unit
vector in the direction of ๐.โข And vice versa!
โข Projected vector: ๐โฅ =๐โ๐
๐๐.
โข The part of ๐ orthogonal to ๐ has no effect!
Cross Product
โข Typically defined only for ๐ 3.
โข ๐ ร ๐ = ๐๐ฆ๐๐ง โ ๐๐ง๐๐ฆ , ๐๐ฅ๐๐ง โ ๐๐ง๐๐ฅ , ๐๐ฅ๐๐ฆ โ ๐๐ฆ โ ๐ .
โข Or more generally:
๐ ร ๐ =
๐๐ฅ ๐๐ฆ ๐๐ง
๐๐ฅ ๐๐ฆ ๐๐ง
๐ฅ ๐ฆ ๐ง
Cross Product
โข The result vector is orthogonal to both vectorโข Direction: Right-hand rule.
โข Normal to the plane spanned by both vectors.
โข Its magnitude is ๐ ร ๐ = ๐ ๐ sin ๐.โข Parallel vectors cross product zero.
โข The part of ๐ parallel to ๐ has no effect on the cross product!
Bilinear Maps
โข Also denoted as โ2-tensorsโ.
โข ๐:๐ ร ๐ โ ๐ , ๐ ๐ข, ๐ฃ = ๐.
โข Take two vectors into a scalar.
โข Symmetry: ๐ ๐ข, ๐ฃ = ๐ ๐ฃ, ๐ข
โข Linearity: ๐ ๐๐ข + ๐๐ค, ๐ฃ = ๐๐ ๐ข, ๐ฃ + ๐๐ ๐ค, ๐ฃ .โข The same for ๐ฃ for symmetry.
โข Can be represented by ๐ ร ๐ matrices: c=๐ข๐๐ ๐ฃ.
Example: Jacobian and Change of Coordinates
โข Suppose change of coordinates ๐ ๐ฅ1, ๐ฅ2, โฏ , ๐ฅ๐ = ๐ฆ1, ๐ฆ2, โฏ , ๐ฆ๐ .
โข Infinitesimal vector ๐ข at point ๐ฅ changes into ๐ฃ: ๐ฃ = ๐ป๐๐ข
โข The bilinear form measures the change in length (stretch): ๐ฃ 2 = ๐ข๐ ๐ป๐โ๐ป๐ ๐ข.
โข Where ๐ป๐โ๐ป๐ is a symmetric bilinear form.โข Which is also a metric.
http://mathinsight.org/image/change_variable_area_transformation
Vector Fields in 3D
โข A vector-valued function assigning a vector to each point in space: ๐: ๐ 3 โ ๐ 3, ๐ ๐ = ๐ฃ.
โข Physics: velocity fields, force fields, advection, etc.
โข Special vector fields:โข Constant
โข Rotational
โข Gradients of scalar functions: ๐ฃ = ๐ป๐.
http://vis.cs.brown.edu/results/images/Laidlaw-2001-QCE.011.html
Integration over a Curve
โข Given a curve ๐ถ ๐ก = ๐ฅ ๐ก , ๐ฆ ๐ก , ๐ง(๐ก) , ๐ก โ [๐ก0, ๐ก1].
โข And a vector field ๐ฃ(๐ฅ, ๐ฆ, ๐ง)
โข The integration of the field on the curve is defined as:
๐ถ
๐ฃ โ ๐ ๐ถ =
๐ก0
๐ก1
๐ฃ โ๐๐ฅ
๐๐ก,๐๐ฆ
๐๐ก,๐๐ง
๐๐ก๐๐ก
Conservative Vector Fields
โข A vector field ๐ฃ is conservative if there is a scalar function ๐ so that for every curve ๐ถ ๐ก , ๐ก โ [๐ก0, ๐ก1]:
๐ถ
๐ฃ โ ๐ ๐ถ = ๐ ๐ก1 โ ๐ ๐ก0
โข Equivalently: if ๐ฃ = ๐ป๐.
โข The integral is then path independent.
Conservative Vector Fields
โข Physical interpretation: the vector field ๐ฃ is the result of a potential ๐.
โข Example: the work (potential energy) ๐ done by gravity force ๐บ =๐ป๐ is only dependent of the height gained\lost.
โข Corollary: the integral of a conservative vector field over a closed curve is zero!
The Curl (Rotor) Operatorโข Definition: ๐ป ร ๐ฃ = ๐ ๐๐ฅ , ๐ ๐๐ฆ , ๐ ๐๐ง ร ๐ฃ.
โข Produces a vector field from a vector field.
โข Geometric intuition: ๐ป ร ๐ฃ encodes local rotation (vorticity) that the vector field (as a force) induces locally on the point.โข Direction: the rotation axis ๐.
โข Integral definition:
๐ป ร ๐ฃ โ ๐ = lim๐ดโ0
1
๐ด ๐ถ
๐ ๐ฃ โ ๐ ๐ถ
โข ๐ถ is an infinitesemal curve around the pointโข ๐ด is the area it encompasses.
http://www.chabotcollege.edu/faculty/shildreth/physics/gifs/curl.gif
Irrotational Fields
โข Fields where ๐ป ร ๐ฃ = 0.โข Also denoted Curl-free.
โข Conservative fields => irrotational.โข as for every scalar ๐:
๐ป ร ๐ป๐ = 0
โข It is evident from the integral definition: lim๐ดโ0
1
๐ด ๐ถ ๐ ๐ฃ โ ๐ ๐ถ.
โข Is irrotational => Conservative fields also correct?
โข Only (and always) for simply-connected domains!
Divergence
โข Definition: ๐ป โ ๐ฃ = ๐ ๐๐ฅ , ๐ ๐๐ฆ , ๐ ๐๐ง โ ๐ฃ.
โข Produces a scalar value from a vector field.
โข Geometric intuition:๐ป โ ๐ฃ encodes local change in density induced by vector field as a flux.
โข Integral definition:
๐ป โ ๐ฃ = lim๐โ ๐
1
๐
๐(๐)
๐ฃ โ ๐
โข ๐(๐) is the surface of an infinitesimal volume around the point.
โข ๐ is the outward local normal.
http://magician.ucsd.edu/essentials/WebBookse8.html
Laplacian
โข The divergence of the gradient of a scalar field:
โ๐ = ๐ป2๐ = ๐ป โ ๐ป๐ .
โข Produces a scalar value from a scalar field.
โข Geometric intuition: Measuring how much a function is diffused or similar to the average of its surrounding.โข Found in heat and wave equations.
โข Used extensively in signal processing, e.g. for denoising.
Stokes Theorem
โข A more general form of the idea of โconservative fieldsโ
โข The modern definition:
๐
๐๐ค =
๐๐
๐ค
โข Geometric interpretation: Integrating the differential of a field inside a domain integrating the field on the boundary.
Stokes Theorem
โข Generalizes many classical results.
โข Integrating along a curve: ๐ถ
๐ป๐ โ ๐ ๐ถ = ๐ ๐ก1 โ ๐ ๐ก0 .
โข Special case: Fundamental theory of calculus: ๐ฅ0
๐ฅ1 ๐นโฒ ๐ฅ ๐๐ฅ = ๐น ๐ฅ1 โ ๐น(๐ฅ0).
โข Kelvin-Stokes Theorem:
๐๐ ๐ฃ โ ๐ ๐ถ = ๐
๐ป ร ๐ฃ ๐๐.
โข Divergence theorem:
๐
๐ป โ ๐ฃ ๐๐ = ๐๐
๐ฃ โ ๐ ๐๐
โข โฆand many similar more.