Vector CalculusA primer

Functions of Several Variables

• A single function of several variables:

𝑓: 𝑅𝑛 → 𝑅, 𝑓 𝑥1, 𝑥2, ⋯ , 𝑥𝑛 = 𝑦.

• Partial derivative vector, or gradient, is a vector:

𝛻𝑓 =𝜕𝑦

𝜕𝑥1, ⋯ ,

𝜕𝑦

𝜕𝑥𝑛

Multi-Valued Functions

• A vector-valued function of several variables:

𝑓: 𝑅𝑛 → 𝑅𝑚, 𝑓 𝑥1, 𝑥2, ⋯ , 𝑥𝑛 = 𝑦1, 𝑦2, ⋯ , 𝑦𝑚 .

• Can be viewed as a change of coordinates, or a mapping.

• We get a matrix, denoted as the Jacobian:

𝛻𝑓 =

𝜕𝑦1

𝜕𝑥1⋮

𝜕𝑦1

𝜕𝑥𝑛

⋯ ⋯𝜕𝑦𝑛

𝜕𝑥1⋮

𝜕𝑦𝑛

𝜕𝑥𝑛

https://www.math.duke.edu/education/ccp/materials/mvcalc/parasurfs/para1.html

Dot Product

• 𝑎, 𝑏 ∈ 𝑅𝑛, 𝑎 ∙ 𝑏 = 𝑎𝑖 ∗ 𝑏𝑖 ∈ 𝑅.

• We get that 𝑎 ∙ 𝑏 = 𝑎 𝑏 cos 𝜃, where 𝜃 is the angle between the vectors.

• Squared norm of vector: 𝑎 2 = 𝑎 ∙ 𝑎.

• Matrix multiplication result dot products of row and column vectors.

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∥ =𝑎∙𝑏

𝑏𝑏.

• The part of 𝑏 orthogonal to 𝑎 has no effect!

Cross Product

• Typically defined only for 𝑅3.

• 𝑎 × 𝑏 = 𝑎𝑦𝑏𝑧 − 𝑎𝑧𝑏𝑦 , 𝑏𝑥𝑎𝑧 − 𝑏𝑧𝑎𝑥 , 𝑎𝑥𝑏𝑦 − 𝑎𝑦 ∈ 𝑅.

• Or more generally:

𝑎 × 𝑏 =

𝑎𝑥 𝑎𝑦 𝑎𝑧

𝑏𝑥 𝑏𝑦 𝑏𝑧

𝑥 𝑦 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 cross product zero.

• The part of 𝑏 parallel to 𝑎 has no effect on the cross product!

Bilinear Maps

• 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

• 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=𝑢𝑇𝑀 𝑣.

Example: Jacobian and Change of Coordinates

• Suppose change of coordinates 𝑇 𝑥1, 𝑥2, ⋯ , 𝑥𝑛 = 𝑦1, 𝑦2, ⋯ , 𝑦𝑛 .

• Infinitesimal vector 𝑢 at point 𝑥 changes into 𝑣: 𝑣 = 𝛻𝑇𝑢

• The bilinear form measures the change in length (stretch): 𝑣 2 = 𝑢𝑇 𝛻𝑇∗𝛻𝑇 𝑢.

• Where 𝛻𝑇∗𝛻𝑇 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: 𝑅3 → 𝑅3, 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 ∈ [𝑡0, 𝑡1].

• And a vector field 𝑣(𝑥, 𝑦, 𝑧)

• The integration of the field on the curve is defined as:

𝐶

𝑣 ∙ 𝑑 𝐶 =

𝑡0

𝑡1

𝑣 ∙𝑑𝑥

𝑑𝑡,𝑑𝑦

𝑑𝑡,𝑑𝑧

𝑑𝑡𝑑𝑡

Conservative Vector Fields

• A vector field 𝑣 is conservative if there is a scalar function 𝜑 so that for every curve 𝐶 𝑡 , 𝑡 ∈ [𝑡0, 𝑡1]:

𝐶

𝑣 ∙ 𝑑 𝐶 = 𝜑 𝑡1 − 𝜑 𝑡0

• Equivalently: if 𝑣 = 𝛻𝜑.

• 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!

The Curl (Rotor) Operator• Definition: 𝛻 × 𝑣 = 𝜕 𝜕𝑥 , 𝜕 𝜕𝑦 , 𝜕 𝜕𝑧 × 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 𝑛.

• Integral definition:

𝛻 × 𝑣 ∙ 𝑛 = lim𝐴→0

1

𝐴 𝐶

𝑑 𝑣 ∙ 𝑑 𝐶

• 𝐶 is an infinitesemal curve around the point• 𝐴 is the area it encompasses.

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: lim𝐴→0

1

𝐴 𝐶 𝑑 𝑣 ∙ 𝑑 𝐶.

• Is irrotational => Conservative fields also correct?

• Only (and always) for simply-connected domains!

Divergence

• Definition: 𝛻 ∙ 𝑣 = 𝜕 𝜕𝑥 , 𝜕 𝜕𝑦 , 𝜕 𝜕𝑧 ∙ 𝑣.

• Produces a scalar value from a vector field.

• Geometric intuition:𝛻 ∙ 𝑣 encodes local change in density induced by vector field as a flux.

• Integral definition:

𝛻 ∙ 𝑣 = lim𝑉→ 𝑝

1

𝑉

𝑆(𝑉)

𝑣 ∙ 𝑛

• 𝑆(𝑉) is the surface of an infinitesimal volume around the point.

• 𝑛 is the outward local normal.

http://magician.ucsd.edu/essentials/WebBookse8.html

Laplacian

• The divergence of the gradient of a scalar field:

∆𝜑 = 𝛻2𝜑 = 𝛻 ∙ 𝛻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:

𝑉

𝑑𝑤 =

𝜕𝑉

𝑤

• Geometric interpretation: Integrating the differential of a field inside a domain integrating the field on the boundary.

Stokes Theorem

• Generalizes many classical results.

• Integrating along a curve: 𝐶

𝛻𝜑 ∙ 𝑑 𝐶 = 𝜑 𝑡1 − 𝜑 𝑡0 .

• Special case: Fundamental theory of calculus: 𝑥0

𝑥1 𝐹′ 𝑥 𝑑𝑥 = 𝐹 𝑥1 − 𝐹(𝑥0).

• Kelvin-Stokes Theorem:

𝜕𝑆 𝑣 ∙ 𝑑 𝐶 = 𝑆

𝛻 × 𝑣 𝑑𝑆.

• Divergence theorem:

𝑉

𝛻 ∙ 𝑣 𝑑𝑉 = 𝜕𝑉

𝑣 ∙ 𝑛 𝑑𝑆

• …and many similar more.