PHY 042: Electricity and Magnetism

Vector calculus: review

Prof. Hugo Beauchemin

Introduction to vector calculus

The aim of the course is to understand Maxwell’s equations Start with their mathematical structures

Need to understand: What vectors are,

Differentiation of vectors fields,

Integration of vector fields,

How to exploit symmetries of a system to simplify the formalization of a problem,

Theorems about vector fields.

Definition of vectors

Vectors are abstract mathematical objects defining linearity from the properties of two operations:

+: (V,V) V

: (S,V) V

E.g.: Will be used to formalize electric and magnetic fields

Properties of these two operations define the set of objects (vectors), that have the structure allowing for linear combinations

Use same concept in various contexts

Can represent samevector in various

ways

Representation of vectors

A subset of all vectors of a given kind can be used to provide a unique decomposition for any other vector of that kind

Basis of a vector space

It can serve as a references for representing a vector in a convenient coordinate system Orthonormal basis

Spectral decomposition

The laws of physics must be independent of the choice of the coordinate system (same physics for any observer) Will exploit this requirement to formulate problems in the

coordinate system that is the most convenient

Special vector functions

Two special functions of vectors (taking two vectors as input): Dot or Scalar product: : (V,V) S

Geometrical interpretation: projection of a vector on another one

E.g.: The components of a vector on the basis of a vector space

Allow to define the norm of a vector

Cross or Vector product: ×: (V,V) V

Geometrical interpretation: area of a parallelogram

Allow to define the determinant of a matrix

Will use the determinant to compute cross product!

Don’t forget the Right-hand rule!!!

Since the cross product yields a vector, we can define two different triple products: Dot product with a cross product

Cross product with a cross product

Derivative Vectors and vector fields can vary in space and in

time Each component can be a function of x, y, z and t

E.g.:

We can differentiate vectors. This changes both the norm and the direction of the vector If we differentiate E with respect to t, then each scalar

functions Ex, Ey and Ez will change, and thus the norm and the direction of the vector field E will change

We know how to differentiate multivariate scalar functions This is vector laden!

Gradient The variation of f between 2 nearby points (x,y,z) and (x’,y’.z’)

is the scalar product of the gradient of f with the displacement vector between these nearby points To be able to differentiate a function, a norm must be defined on

the vector space

The gradient is a vector It has a norm and a direction

Interpretation: Point in direction of biggest increase of f (x,y,z)

Its norm gives the rate of the increase along the largest increase direction

Null variation of df: Equipotential

Stationary points

Differential Operator Recall: vectors are not arrows but abstract entities satisfying

linearity operations

We can define a vector differential operator:

Define vector differential operator in 3D as:

o Is there a difference between and ?

Q: Why do we speak of a “vector operator”?

A: It is an object that acts differently on each component of a vector to transform these components according to a well defined operation.

Allows to define the derivative of multivariate vector fields!

Divergence The divergence of a vector field is a scalar!

It is a number telling how much a vector is spreading out from a point, i.e. how much the lines of the field diverge from this point

Note: These are not picture of the divergence of a field but picture of fields with different divergences.

Curl The curl of a vector field is a vector!

The norm tells you how much the field is curling around an axis at a given point

The direction tells you around which axis the original field is curling

Use the right-hand rule to get this axis

Sum and product rules Sum rule:

The derivative of a sum of vector is given by the sum of the derivative of the elements of the sum

Thanks to the linearity of the vector differential operator and of the vectors summed

Product rule: many possibilities The product of two functions (vector or scalar) to be differentiated can

be:

Scalar: or

Vector: or

The derivative of a scalar function is a vector

Gradient

The derivative of a vector function is a:

Scalar: the divergence

Vector: the curl

6 POSSIBILITIES

Second derivative Equation with 2nd derivative are often used in

physics Field equations in function of potential such as

Poisson

Characterize an extremum

There are 5 possibilities of second derivativeGradient of a scalar Vector

Divergence

Curl

Divergence of a vector Scalar Gradient

Curl of a vector VectorDivergence

Curl

Integral vector calculus I Problem to solve:

We want to perform definite integrals (integrals with specific boundary conditions) of vector fields

Generalize the Fundamental Theorem of Calculus (FTC)

Boundaries: The boundaries a and b enclose, limit, the integral

interval. If the integral cover many dimensions (or just 1D but embedded in an higher dimensional space), boundaries can be defined as:

Line (path) integral

Surface integral

Volume integral

Integral vector calculus II

Path integral: An infinitesimal element along the path on which the integral is to be taken

Note: is constrained to be on the path so

Can be parameterized by one variable (e.g. time t)

Close Integral From the FTC, if a=b, the integral is null,

need not be the case if the integral is over a close path

Example? Work done on a system by a non-conservative force such as friction

Integral over a path (1D)

To integrate a vector function over a given path P, we need:

1. Parameterize the path as a 1 variable function

2. Project the vector function onto the path

3. Integrate the product

A

B

Only the lines of field along the path matter

Only the components of field parallel to the path matter

Conservative forces A force is conservative if the work it does along a

path P is independent of that path, and only depends on the starting and ending points (path integral of F just depends on a and b)

If the vector field to integrate is a gradient, then this vector field describes a conservative force This is just a special case of vector function

P

P’

Conservative Non-Conservative

-Independent of P

-W=W’

-

-E.g. Falling stone

-Depends on P

-W≠W’

-

-E.g. Pushing a box on the ground

Surface integrals (2D) What if the function is to be integrated over a 2D

surface rather than a 1D path? Need to define an infinitesimal patch of area

constrained to the surface to integrate over and compute

Q: How to define a vector for a surface element da ?

A: The direction is set by the normal to the patch area

Provide a measure

of the amount of F

that flows through S

Conventions:• S is closed: da is outward to the enclosed surface• S is bounded by C: da obtained by right-hand rule

A question:

Q: What is the difference between the

circulation of a

vector field and the flux of the vector field ,

i.e. the

difference between and

when the

surface S is formed by a closed curve C?

A: The circulation is the integration of the tangential

component of the vector field on C and the flux is

the

integral of the normal component to C.

Flux-divergence theorem

I claimed in intro that the formulation of Maxwell’s equations in terms of curl and divergences are equivalent to their formulation in terms of flux and circulation

This is established by theorems

Flux-divergence (a.k.a. Gauss or Green theorem): The integral of a derivative over a volume is equal to

the primitive at the boundary, i.e. the whole surface enclosing that volume

Flux across a surfaceMeasure of the outgoing divergence of a field from a point per unit volume

Stokes theorem Another version of the fundamental theorem of

calculus: The integral of a derivative over a surface is equal to

its primitive at the boundary of the surface, i.e. over the circulation enclosing the surface

Circulation of the field at theboundary

Measure of the total amountof “swirl” of a vector fieldinside an area S

Spherical coordinates Vectors are independent of coordinates

(same vectors in different bases) Can exploit the symmetry of a system to

simplify the problem

The price to pay is to have a reference system that changes from point to point in the space

Cylindrical coordinates Useful when you have a field generated by a wire, or

by a solenoid for example ATLAS/CMS detectors have cylindrical symmetry

Like spherical coordinates but with q=p/2

Helmholtz theorem Generally, in E&M, we don’t know what the electric

and magnetic fields are, but we know how they vary Differential equations (Maxwell’s equation)

Is the divergence and the curl of a vector field determine the field at every point of space where these derivatives are defined and known?

Thm: A field is uniquely determined by its divergence, its curl (in a finite volume) and its boundary conditions

Charge source Current source

E&M: Use experiments to find relationships between sources and field variations

Potentials Under certain conditions, fields can be obtained from the

knowledge of the variation of underlying functions called potentials

Theorem 1: If the curl of a vector field is null: The circulation of the field over a closed path is null The field is a conservative force The field can be expressed as the gradient of a scalar

potential

Theorem 2: If the divergence of a field is null: The flux is null over a closed surface The field can be expressed as the curl of a vector potential

For ANY field (always!):