div, grad, curl, and all that - a review

∇·, ∇, ∇×, and all that
a review by Kwanghee Choi

Reference: Everything is based on this book
Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Fourth Edition)
by H. M. Schey

Chapter 1
Introduction,
Vector Functions,
and Electrostatics

1.1 Prerequisites: Vector Functions
- Vector function F specifies a magnitude and a direction at every point.

1.1 Prerequisites: Vector Functions
- Vector function can be resolved into components.

1.2 Prerequisites: Simple Electrostatics
- Much of the vector calculus was invented for use in electromagnetic theory:
What vector calculus is, What vector calculus is for
- Context not exclusively mathematical - for physical and geometric intuition
- Three things you should know:
- Existence of electric charge
- Coulomb’s law
- Principle of superposition

1.2.1 Electric charge
- Two kinds of charge: positive and negative.
- Every material body contains electric charge.

1.2.3 Principle of Superposition
- Force between two charged particles is not modified by the presence of other charged
particles.
- If F1
is the force exerted on q0
by q1
, and F2
by q2
, then the net force is F1
+ F2

1.2.4 Electrostatic Field
- Force per unit charge
- Vector function of position
- Region of space in the vicinity of a charge
- Principle of superposition can be applied.
- Electromagnetism is a field theory.

Chapter 2
Surface Integrals
and the Divergence

2.1 The Unit Normal Vector
- “Normal” means “perpendicular” in this context.
- “Unit” means its length is 1.

2.2 Definition of Surface Integrals

2.2 Definition of Surface Integrals
- Let G(x, y, z) a given scalar function, rather than the dot product of two vectors.
- Approximate surface S by a polyhedron, sum over, and then take the limit.

2.2.1 Direction of the normal vector
- Case of closed surface:
Points outward the enclosed surface (“A gentlemen’s agreement”)
- Case of open surface:
Have to be given explicitly

2.3 Evaluating Surface Integrals

- Latin for “flow”
2.4 Flux

- The electric field “flows” out of a surface enclosing charge.
- “Amount” of this “flow” are proportional to the net charge enclosed.
2.4.2 Gauss’ Law

- Suppose we are doing a numerical calculation of Gauss’ Law.
- 10 unknowns.
- 100 unknowns.
- Infinitely many unknowns.
- It is far more easier to deal with the “flux at a single point”,
rather than the “flux through a surface”.
2.4.3 Use Gauss’ Law to Find the Field

- “Flux at a single point”
- Consider the density ρ of electric charge at a single point (x, y, z).
2.5 Divergence

- “Flux at a single point”
- Density ρ of electric charge at a single point (x, y, z)
- The field E “diverges” from a point.
How much it “diverges” depends on how dense charge is.
2.5.3 Interpretation of Divergence

- Surface integrals from the common face S0
cancel each other,
therefore does not contribute to the sum of each box’s total flux.
2.6 The Divergence Theorem

- Expression of a conservation law.
- Basic equation for electromagnetic, hydrodynamics and diffusion theory.
- Proof omitted.
2.6.1 Continuity Equation

Chapter 3
Line Integrals
and the Curl

- Work done in one dimension, if a force F(x) acts from x = a to x = b.
3.1 Work and Line Integrals

- Work done in three dimensions, if a force f(x, y, z) acts along the curve C.
3.1 Work and Line Integrals

- Let t denote a unit vector that is tangent to the curve at P.
3.2 Line Integrals Involving Vector Functions
^

- t denote a unit vector that is tangent to the curve at P.
^

- Under some conditions, the line integral does not depend on the path.
- In the case of Coulomb force,
we’d get the same answer for any path connecting P1
and P2
.
3.3 Path Independence

3.3.1 Line Integral around a Closed Path

- What if the line integral is path dependent?
- In other words, what if line integral around a closed path is nonzero?
3.4 The Curl

- Curl has something to do with a line integral around a closed path.
- Somehow has to do things with rotating, swirling, or curling around.
- Path dependence around a closed path surrounding infinitesimal area, i.e. a point.
- Nonzero work by rotating a paddle on a point.
3.5 The Meaning of Curl

- Consider some closed curve C and a capping surface S.
- Approximate S by a polyhedron of N faces,
automatically creating a polygon P that approximates C.
3.6 Stokes’ Theorem

3.6 Stokes’ Theorem
- Line integrals from the common segment AB cancel each other,
therefore does not contribute to the sum of each faces’ total line integrals.

3.6.1 Differential Form of Ampère’s Law
- Special case of Maxwell’s equations, valid when the fields do not vary with time.
- Proof omitted.
- Ampère’s Law
- Stokes’ Theorem
- Differential Form of Ampère’s Law

3.6.2 Simply Connected Region
- For the Stokes’ theorem to hold throughout some region D,
for any closed curve C lying entirely in D,
its any capping surface also has to lie entirely in D.
- We refer to those regions as simply connected.
- Torus is not a simply connected region,
as the capping surface cannot lie entirely in D for a given closed curve C.

4.1 Line Integrals and the Gradient
- Suppose that a given vector function F(x, y, z) has associated with
a scalar function Ψ(x, y, z) with the following relations.
- Let us show the equivalence between the preceding relation,
and the line independence of the line integral of F·t, or namely,^

- First, let us assume that the preceding relations hold.

- The result of line integral only depends on (x1
, y1
, z1
) and (x0
, y0
, z0
).
- Therefore, it is path independent.

- Conversely, let us assume that line integral is independent of path.
- Let us choose partial path C1
as a straight line with y and z as constant.

- Choose paths for y and z similarly.
- The results are the same with the relations we were aiming to derive.
- Therefore, we have proven the equivalence.

- Solid arrows represent implications
that hold in general,
provided that F is smooth.
- Dashed arrows represent implications
that hold if F is smooth,
and region of interest has to be
simply connected.

4.2 Directional Derivatives and the Gradient
- One-dimensional taylor series says that the value of the function at some point x + Δx
can be written as the sum of infinitely many terms that involve the function and its
derivatives at some other point x.
- Taylor series can also be formed for functions of several variables.

- df/ds is called the directional derivative of f,
which is the rate of change of the function f in the direction of Δs.
- Results all apply to functions F of three or more variables.
Δs

- Maximum dF/ds is at α = 0.
- The gradient of a scalar function F(x, y, z) is a vector
that is in the direction in which F undergoes the greatest rate of increase
and that has magnitude equal to the rate of increase in that direction.
4.3 Geometric Significance of the Gradient

- If the dot product of two vectors, neither of the zero vanishes,
the two vectors are perpendicular.
- ∇T is normal to the surface T = const.

- Any displacement vector s from the surface f = const can be split into
two scalar components, along the surface s||
and one normal to it s⊥
.
- Only the normal component carries us away from the surface
and causes a change in f.

- For any closed path C, the electrostatic field E can be written as the gradient of a
scalar function, called the electrostatic potential Φ.
- By combining it into the differential form of Gauss’ law,
- we get the Poisson’s equation.
4.4 Electrostatic Potential

4.4.2 Geometric Interpretation of the Poisson’s Equation
- Since ∇Φ is a vector in the direction of increasing Φ,
the force on a positive charge q is F = qE = -q∇Φ,
which is in the direction of decreasing Φ.
- This is the reason for the negative sign in the definition of electrostatic potential.
The negative sign ensures that a positive charge moves “downhill”
from a higher to a lower potential.

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