Vector Calculus Subject Notes

Space curves and vector fields

Introduction to vector fields A vector field is an assignment of a vector to each point in a subset of space. A vector field in the

plane can be visualised as a collection of arrows with a given magnitude and direction, each attached

to a point in the plane.

By contrast, a scalar field has only one component.

Example:

Example:

Parametric curves The position at time along an arbitrary curve in can be represented by the path:

Example: Consider the curve parameterised as follows.

The parameter defines a unique point along the curve for the and components. This gives

rise to the following definitions:

The tangent line to the path at is:

Arclength

The arclength of a path for is given by adding up all the

incremental changes in position along the path. Hence we can write:

Taking the limit as becomes infinitesimal we find:

Normal and tangent vectors The unit tangent vector to a path is just the normalised velocity vector:

The unit normal vector is closely related to the acceleration vector:

The curvature of a path can be measured as the magnitude of the normal vector relative to the

magnitude of the velocity vector:

Differentiation in Scalar and Vector Fields

Differentiability in fields The function is differentiable at if:

To be continuous, the following limit must be defined as approaches along any path in

the domain:

A function is said to be if all of its th order partial derivatives exist and are continuous.

Matrix derivatives

The partial derivative of a vector field with respect to its single parameter is

written using the chain rule as follows:

This approach can be expanded to the case of a vector field

whose components depends on three parameters :

This can be written in matrix form. If and are differentiable functions and

the composition is defined, then the set of partial derivatives is defined as follows:

Here is a dimenstional matrix and is a dimensional matrix, resulting in

being a dimensional matrix.

If , , and (the full three dimensional case) then this can be written in full as:

In the last three lines we have represented as a column vector to simplify the notation. We can

now see that, given that we have made the substitution , , , the matrix results

match the results we obtain by using the iterated chain rule without matrix notation.

Now take the simpler , , and (two dimensional case) we find:

Taylor polynomials Any function that can be differentiated across a domain up to times can be approximated by a

polynomial of order about a point . These are called Taylor polynomials.

The first-order Taylor polynomial for a bivariate function can be written as:

In this last line we have written everything in vector form, with and .

In the general th order case this becomes:

Example: find the 2nd order Taylor polynomial for near the point .

Extrema

The critical points of a function are those points where , or where does not exist.

Critical points correspond to either local maxima, local minima, or saddle points. Which case it is can

be determined by calculating the determinant of the Hessian matrix, the matrix of partial

derivatives.

The possible outcomes are as follows:

Case Type of critical point

and Local maximum

and Local maximum

Saddle point

Inconclusive test

Differential operators in vector fields The following table outlines the four main differential operators of relevance for vector/scalar fields.

Name Symbol Definition Geometrical Interpretation

Gradient

Vector pointing in the ‘uphill’ direction of the field at a given location

Divergence

Magnitude of how much the field points ‘outwards’ at any given location

Curl

Vector pointing along the axis of rotation, with magnitude proportional to strength of rotation at a given point

Laplacian

Magnitude of how much the slope is changing at any given point

Note: the gradient of a vector field is calculated component-wise, and produces a 3x3 tensor. The

Laplacian of a vector field is a 3-length vector.

Example: illustration showing the gradient vector on the level curves of a scalar field.

Example: illustrative vector fields with positive, negative, and zero divergence.

Example: a vector field rotating counterclockwise about the z-axis, resulting in a curl vector along

the positive z-axis as determined by the right hand rule.

Differential operator identities The curl of a gradient is zero

This states that the rotational component of the slope of a scalar field is equal to zero:

This can be proved as follows.

Since by assumption is twice differentiable, the order of partial differentiation does not matter.

Hence we have:

A scalar field with no curl component is known as irrotational.

The divergence of a curl is zero

This states that the rotational component of a vector field has no outward-projecting component:

This can be proved as follows.

Since by assumption is twice differentiable, the order of partial differentiation does not matter.

Hence we have:

Other identities

Property Identity

Linearity of grad under addition

Linearity of grad under scalar multiplication

Product rule for grad

Quotient rule for grad

Linearity of div under addition

Product rule for div

Linearity of curl under addition

Product rule for curl

Product rule for laplacian

Distributive rule for divs

Divergence of cross product of divs

Fundamental Theorem of Vector Calculus Combining the two main theorems discussed above, we can determine that any differentiable vector

field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field

and a solenoidal (divergence-free) vector field ; this is known as the Helmholtz decomposition.

We call the scalar potential and the vector potential of the field .

Integration in Scalar Fields

Two-dimensional Cartesian coordinates A double integral is the volume under the surface of a bivariate scalar function that lies

above the domain in the plane.

Fubini’s Theorem states that, for most well-behaved functions, a double integral can be computed as

an iterated integral, where the order of integration does not matter.

When swapping the order of integration, the boundaries need to be adjusted so that they are

written in the correct variable. This usually requires sketching the area to be integrated. In the case

below, the boundaries could be written in two ways:

Example: evaluate the following double integral.

Three-dimensional Cartesian coordinates A triple integral is calculated in the same way as a double integral, but with an additional coordinate.

The physical interpretation changes depending on the application.

The challenge with triple integrals is in determining the correct terminals of integration these

depend both on the order of integration and on the volume being integrated over. For anything but

the simplest shapes it is usually necessary to draw a diagram.

To find the average value of a scalar field over a domain, the total value of that scalar field over the

domain can be divided by the volume of that region:

Example: solve the integral of over the following domain .

To determine the limits of integration, begin with varying from to , then project the equation

for the plane into the plane by setting , and solving for . Finally, simply rearrange

for . Putting this together we find:

Example: find the volume of the following shape.

To determine the limits of integration consider each axis in turn:

x y z

For For

Now we can write the integral:

Use

Using the results:

We can substitute back in for :

Changing coordinate systems Change of variables

Where is the Jacobian matrix defined as:

Change of variables

Where is the Jacobian matrix defined as:

Cylindrical coordinates Spherical coordinates have the following parameterisation:

To perform the transformation of the integral we need to calculate the Jacobian:

We therefore have the conversion:

Example: find the volume of the following shape.

To determine the limits of integration consider each axis in turn:

Use symmetry Set Use equation

Now we can write the integral:

Spherical coordinates Spherical coordinates have the following parameterisation:

To perform the transformation of the integral we need to calculate the Jacobian:

We therefore have the conversion:

Example: find the volume of a radius R sphere by integration.

To determine the limits of integration consider each axis in turn:

Use symmetry Use symmetry Use equation

Now we can write the integral:

This is the well known formula for the volume of a sphere.

Integration in Vector Fields

Line integrals Integral of scalar field along the path :

We can generalise this result to calculate the integral of a vector field along the path :

Surface integrals Consider a differentiable surface , and two curves and on parameterised with coordinates

and . The paramaterisation of the surface can be written as:

is the tangent vector to curve on at point .

is the tangent vector to curve on at point .

The cross product of these two tangent vectors defines a vector that is normal to the surface at the

point .

The magnitude of this vector represents the area of the parallelogram spanned by the two normal

vectors and , such that we can write:

In the limit of small increments along each tangent curve this becomes:

Note that:

The function we are summing over the area is equal to the outward component of the vector field,

which is written in terms of the unit normal vector as:

The integral of all such flux over the defined area can therefore be written as:

This formula is most useful if we already know the unit normal vector. If we don’t, we notice the

definition of the unit normal:

Substituting this into the formula above yields:

This integral is typically written using the differential notation:

One special application of surface integrals is to calculate the volume over some region by

calculating the surface integral over the surface over the vector field .

Example: find the flux of the field across the parabolic cylinder surface

.

This surface can be parameterised as . Hence we have:

Volume integrals While in the section on integration in scalar fields we focused mostly on volume integrals, which are

solved by computing a triple integral, in vector fields this operation is seldom useful. Instead

integration in vector fields is concerned mostly with performing line and surface integrals over

vector fields, and understanding the relations between these quantities. This leads us to a discussion

of some of the key theorems pertaining to integration in vector fields.

Green’s Theorem in the Plane Consider a region in the plane , with the boundary of that region and the orientation of

that surface specified by the right-hand rule as indicated in the diagram below.

Green’s Theorem in the plane states that at the line integral around the boundary is equal to the

sum of the curl throughout that region. This is written as follows:

Or in more general vector form:

This can be visualised as saying that the line integral about a region is equal to the sum of the

infinitesimal rotational elements around the boundary.

Example: evaluate the line integral over as defined below, over the unit circle centred at the

origin in .

Since the line integral is difficult to evaluate, we will use Green’s Theorem in the plane.

Use polar coordinates to solve:

Divergence Theorem in the Plane The divergence theorem states that for a domain in the xy plane, the integral of the divergence of

a vector field over that domain is equal to the line integral of the outwards normal component of

that field over the boundary of the domain. This is written:

Stokes’ Theorem For a surface and a closed boundary of that surface , then the sum of the curl of a vector field

over the surface is equal to the line integral over the boundary of that surface. This is written:

Stokes’ Theorem is thus a generalisation of Green’s Theorem in the plane.

Example: Evaluate the surface integral of as defined below, over the for

.

This surface integral is too hard to solve, so evaluate the line integral using Stokes’ Theorem.

We can parameterise the curve as

.

This integral is also too hard to solve. However, Stokes’ Theorem says that the surface integral only

depends on the values of the field at the boundary, so we can find a simpler surface that has the

same boundary, and evaluate the integral this way.

We use the paramaterisation of the circle

By Stokes’ Theorem, since (i.e. they share a boundary), the surface integrals will be equal

to each other, and hence:

Note that if a surface is closed, Stokes’ Theorem implies that:

Gauss’ Divergence Theorem Consider a solid closed region in denoted , and a closed oriented surface bounding that

closed region. Gauss’ Divergence Theorem states that the integral of the divergence of a vector field

throughout the region is equal to the integral of that vector field across the boundary of that

region. This is written:

This can be visualised by considering the divergence from many tiny units of volume inside the

region . The divergence out of one will offset that of the divergence out of neighbouring units for

all volume units apart from those on the surface, which is the same as the surface integral.

Example: evaluate the surface integral of over the

surface , given by .

If we incorporate the bottom surface of the shape , the surface becomes

closed and we can use Gauss’ Divergence Theorem.

From this we can deduce that:

We will solve using polar coordinates, with gives a surface parameterisation of:

Using this parameterisation and the formula for a surface integral with known we have:

Conservative Fields A vector field is said to be conservative if the integral over any path in that field is independent of

the form of the path, but depends only on the endpoints of the path.

A vector field will be conservative if it is irrotational, meaning that it has no curl, and hence can be

written as the gradient of a scalar function. Thus:

Hence we see that the integral only depends upon the value of the field at the start and endpoints.