Vector Calculus

Introduction to the course. What is vector calculus about?

Quick refresh of vectors: notation, canonical basis, components, length, direction, unit vectors.
Vectors as elements of R³, as triples of real numbers, as points in space, as geometric objects with direction and length.

1.1

Operations with vectors. Vector addition and scalar-vector multiplication.

Scalar product: definition, use of scalar product to compute angles, magnitudes and lengths of projections, parallel and orthogonal vectors.

Vector product: definition, geometric interpretation, anticommutativity, non-associativity.
Triple product: definition, volume of a parallelepiped.

Open sets, definition, intuitive description as sets containing spheres centred at each point.

1.1.1,

1.1.2,

1.1.3

10.2 (p.577)
Ex. 2-3, 16-31

10.3 (p.584)
Ex. 1-28

Limits of sequences of vectors. Closed sets: definition, intuitive description as sets containing limits.

Scalar fields: definition, definition of continuous fields, level sets, representation of two-dimensional fields with graph surfaces, example f=x²+y².

Vector fields: definition, quiver representation, examples (F=i+yj and F=i+xj).

Curves: definition, path of a curve, example a(t)=cos(t)i+sin(t)j.

1.2.1,

1.2.2,

1.2.3

12.1 (p.676)
Ex. 11-42

15.1 (p.848)
Ex. 1-8

C2,
C3,
C4,
C5

Recall definitions of curve, path and loop. Interpretation of a curve as trajectory of a moving point. Different parametrisations of the same path.
How to draw the path of a curve? Example: circle.
How to compute the parametrisation of a path? Examples: segments and graphs (Remark 1.24).

Partial derivatives: definition, interpretation, examples. Linearity, product rule and chain rule.

Gradient: definition, examples, linearity, product rule and chain rule.
Directional derivatives.

1.3.1,

1.3.2

11.3 (p.641)
Ex. 1-12

12.3 (p.687)
Ex. 1-12, 25-31

12.7 (p.723)
Ex. 1-6 (part a),
10-15

C.6

The gradient of f is perpendicular to the level lines of f.
The gradient of f points in the direction of steepest ascent of f.
Examples: the field f=x²+y² and the "height above sea level" scalar field.

Jacobian matrix of a vector field.

Second order partial derivatives, Hessian of a scalar field, Laplacian, harmonic functions. Clairaut's Theorem, examples.

Divergence of a vector field. Definition, examples, physical interpretation.

1.3.3,

1.3.4,

1.3.5

12.4 (p.692)
Ex. 1-14

16.1 (p.896)
Ex. 1-8

C.7,
C.9

Exercises C.2, C.3, C.4.

Curl of a vector field. Definition, example, interpretation as rotation.
Summary of the differential operators encountered so far.

Vector differential identities: second-order vector differential operators, vector product rules. Proof of identities (24), (26), (29) and (31).

1.3.6,

1.4.1,

1.4.2

16.2 (p.902)
Ex. 8, 11, 13, 14

C.8, C.10,
C.11

Two proofs of Laplacian's product rule (32).

Special fields: definition of irrotational, solenoidal and conservative fields, scalar and vector potentials.
Non-uniqueness of potentials.
Minimal examples of fields which are only solenoidal, only irrotational, both or none (yi, xi, i or xi-yj, (x+y)i).
A field is irrotational if and only if its Jacobian is symmetric.
Conservative implies irrotational, existence of vector potential implies solenoidal; what about the converse? it depends on "holes" in the domain (no details on this).
Every 1D function is derivative of another one but not all vector fields are gradients or curls.

How to compute potentials? Integration of a differential equation; example: computation of the scalar potential for xi+yj.

1.5

15.2 (p.857)
Ex.1-10

16.2 (p.902)
Ex. 9, 10, 12, 15, 16

C.12

Exercises C.5, C.6, C.7.

Example: compute vector potential for -yi+xj.

Total derivatives of curves. Linearity and product rules for derivatives of curves.
Total derivative is tangent to the path of a curve; its magnitude is velocity. Intrinsic parametrisations and arc length.

Recall chain rules already seen: for composition of real functions and for composition of a real function with a scalar field.
Chain rule for the evaluation of a scalar field along a curve.
Example: total derivative of the field f=xy constrained to the curve a(t) = cos t i + sin t i; two solution procedures.

1.6,

1.7

11.1 (p.627)
Ex. 1-14, 27-32

C.13,
C.14,
C.15

Chain rule for the partial derivatives of the composition of a scalar field with a vector field.
Example 1.85: derivative of a scalar field constrained to a graph surface.

Summary of the programme covered so far, structure of chapter one of the notes.

General introduction to vector integration.
Interpretation of integrals as measures (lengths, areas, volumes) of geometric objects or as computation of a mass from a given density.
Three fundamental properties of integrals: linearity, additivity, integral of constant 1 is measure of domain of integration.

Line integrals of scalar fields: formula for the length of a path and for the line integral of a scalar field.

2.1.1

12.5 (p.702)
Ex. 1-12, 15-22

C.16

Exercises C.8, C.9, C.10, C.12.

Recall line integrals of scalar fields and their interpretation.
Measure of the length of a curve and of the graph of a scalar function.
Independence of parametrisation.
Example: computation of line integral of f=y³ along two different parametrisations of half circle.

Line integrals of vector fields: formula, unit tangent vector, interpretation of line integral as integral of tangential component and relation with integral of scalar fields.
Example: integral of F=r over segment from i to i+j, and over the same segment in the opposite direction.
Oriented paths, changing path orientation changes sign of the integral.
Circulation, notation for integrals over loops.

2.1.2

11.3 (p.641)
Ex. 13-21

15.3 (p.861)
Ex. 1-9

15.4 (p.869)
Ex. 1-7

C.17

Recall line integrals of scalar and vector fields and their interpretation.
Other kind of line integrals: int_Gamma f dx.

Recall fundamental theorem of calculus (for real functions).
Fundamental theorem of vector calculus; example.
Path independence of the line integral of conservative fields.
Scalar potentials as primitives for line integrals.

Theorem (only stated): given a vector field F, three conditions are equivalent: (i) F is conservative, (ii) loop integrals of F are zero, (iii) line integrals of F are path-independent (for paths with the same endpoints).

Definition of star-shaped domain.
Theorem: given vector field F defined on a star-shaped domain, F is conservative if and only if it is irrotational.

2.1.3

Justification of formula for the line integral for scalar fields using Riemann sums (on equispaced points).

Exercises C.13 and C.14.

6

...

Exercises C.15 and C.16.

Double integrals: iterated integral on y-simple regions, geometric interpretation as volume between graph and xy-plane, comments on more general regions. Example: integral of xy on rectangle (0,2)x(0,1) and on the triangle with vertices 0, 2i and 2i+j.

Change of variables for double integrals: motivation from integration by substitution in 1D, new coordinates xi and eta, vector field T, xi-eta-plane, unit vectors, Jacobian and Jacobian determinant, change of variables formula.

2.2.1,

2.2.2

14.1 (p.796)
Ex. 13-22

14.2 (p.802)
Ex. 1-28

14.4 (p.817)
Ex. 32-34

C.18,
C.19,
C.20,
C.21

Double integral example: integral of (x+y)¹² over the square with vertices i, j, -i, -j.

Triple integrals, z-simple domains, volume of a domain, change of variables formula.

Comparison: path integrals extend scalar integrals from "flat" to curvilinear (1D) domains of integration.
Similarly, extending double integrals from "flat" to curvilinear (2D) domains we obtain surface integrals.
Surfaces as two-dimensional curved objects in R³. Definition of parametric surfaces

2.2.3,

14.5 (p.823)
Ex. 1-20

Recall definition of parametric surfaces.
Example: parametrisations of paraboloid {z=x²+y²} and cylinder {x²+y²=1}.
Graph surfaces as parametric surfaces.
Tangent and orthogonal vectors on a surface.

Formula for the integral of a scalar field over a parametric surface.
Independence of parametrisation.
Special case: integral of a scalar field over a graph surface.

2.2.4

14.7 (p.838)
Ex. 1-10

15.5 (p.880)
Ex. 3, 4, 7-10, 13-16

Example: integral of f=√(1+4z) over the paraboloid defined by X=ui+wj+(u²+w²)k over R=(-1,1)².

Flux of a fluid flowing through a surface.
Unit normal vector fields, orientations and oriented surfaces. Non-orientable surfaces (Moebius strip).
Parametric surfaces (including graphs), boundaries and level sets are orientable and have a preferred choice of orientation.

Flux as surface integral of normal component of vector field; analogy with line integrals.
Formula for fluxes on a parametric surface and on a graph surface.
Example: flux of F=-yi+xj+zk through the paraboloid defined by X=ui+wj+(u²+w²)k over R=(-1,1)².

Path orientation induced by an oriented surface on its boundary.

2.2.5

15.6 (p.886)

C.22

Exercises C.18, C.19, C.20

Special coordinate systems are standard changes of variables used in presence of some geometric symmetries.

Polar coordinates r and ϑ. Definition, Jacobian determinant and area element, area of a star-shaped region, disc, orthogonal unit vector fields r and ϑ.

Cylindrical coordinates r, ϑ and z. Definition, volume element, volume of a solid of revolution.

Spherical coordinates ρ, Φ and ϑ. Definition, volume element.

2.3.1,

2.3.2,

2.3.3

8.5 (p.488)
Ex. 1-28

8.6 (p.492)
Ex. 1-11

10.6 (p.600)
Ex. 1-14

14.4 (p.817)
Ex. 1-30

14.6 (p.830)
Ex. 1-16

15.2 (p.857)
Ex. 19-22

C.23,
C.24,
C.25

Introduction to new part: three theorems on integration of derivatives.
Recall fundamental theorem of calculus and fundamental theorem of vector calculus.
Possible generalisations of the different objects appearing in these theorems.

Green's theorem: assumptions, statement, comments, derivation from Lemma 3.3.
Proof of part 1 of Lemma 3.3.

3.1

Table of comparison of the different integrals considered so far.

Exercise C.21.

Completion of the proof (parts 2 and 3) of Lemma 3.3.
Use of Green's theorem to compute areas.

Exercise: compute circulation of F = (x-y)i + xj around a circle or a triangle in the xy-plane.
Exercise: for a smooth real function g, prove that the circulation of a vector field F = g(x+y)(i + j) around the boundary of any planar region is zero.

Divergence theorem: assumptions, statement, interpretation.

C.26

Recall divergence theorem. Proof using Lemma 3.13.
Proof of Lemma 3.13.

Divergence of a vector field as average flux through infinitesimal spherical surface.

3.2

16.4 (p.912)
Ex. 1-14

C.27,
C.28,
C.29

Comments on errors found in the assignments.
Exercise C.22.

Integration by parts in three dimensions, derivation of formula in 3.26.
Derivation of first and second Green's identities (Corollary 3.22).

Stokes' theorem.
Statement and simple consequences.

"Physical" definitions of divergence as (averaged) flux through an infinitesimal sphere and of curl as (averaged) circulation around an infinitesimal disc (Remark 3.33).

Recall relation between conservative fields and path independence of line integrals (box (50)); similar relation between fields admitting a vector potential and "surface independence" of fluxes (Remark 3.34).

Example of application of the divergence theorem: radial fields that are solenoidal outside a ball decay as 1/r²; for example, gravitational and electrical fields.

3.3

16.5 (p.916)
Ex. 1-12

C.30,
C.31,
C.32,
C.33

Example of application of the divergence theorem to physics: derivation of continuity equation for a fluid in absence of sources and sinks.

Exercises C.27, C.24, C.26.

Exercises C.28 and C.30.

Exercises 1, 2 and 4 of the 2016 MA2VC exam.