1. PHY 042: Electricity and Magnetism
Vector calculus: review
Prof. Hugo Beauchemin

2. 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.

3. Definition of vectors
Use same concept
in various contexts
Can represent same
vector in various
ways
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

4. 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

5. 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

6. 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!

7. 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

8. 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:
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!

9. 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.

10. 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

11. 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

12. 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 derivative
Divergence
Gradient of a scalar  Vector
Curl
Divergence of a vector  Scalar
Gradient
Divergence
Curl of a vector  Vector
Curl

13. 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

14. 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

15. Integral over a path (1D)
To integrate a vector function over a given path P, we need:
Parameterize the path as a 1 variable function
Project the vector function onto the path
Integrate the product
Only the lines of field
along the path matter A
Only the components of
field parallel to the path
matter B

16. 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
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 P P’

17. 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

18. 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.

19. 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
Measure of the outgoing
divergence of a field from
a point per unit volume
Flux across a surface

20. 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
Measure of the total amount
of “swirl” of a vector field
inside an area S
Circulation of the field at the
boundary

21. 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

22. 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

23. 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

24. Potentials For ANY field (always!):
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!):