Summary of the book "div, grad, curl, and all that" Covers divergence, gradient, curl, and related theorems.

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

2. 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

3. Chapter 1
Introduction,
Vector Functions,
and Electrostatics

4. 1.1 Prerequisites: Vector Functions
- Vector function F specifies a magnitude and a direction at every point.

5. 1.1 Prerequisites: Vector Functions
- Vector function can be resolved into components.

6. 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

7. 1.2.1 Electric charge
- Two kinds of charge: positive and negative.
- Every material body contains electric charge.

8. 1.2.2 Coulomb’s law

9. 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

10. 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.

11. Chapter 2
Surface Integrals
and the Divergence

12. 2.1 The Unit Normal Vector
- “Normal” means “perpendicular” in this context.
- “Unit” means its length is 1.

13. 2.1 The Unit Normal Vector

14. 2.1 The Unit Normal Vector

15. 2.2 Definition of Surface Integrals

16. 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.

17. 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

18. 2.3 Evaluating Surface Integrals

19. 2.3 Evaluating Surface Integrals

20. 2.3 Evaluating Surface Integrals

21. 2.3 Evaluating Surface Integrals

22. - Latin for “flow”
2.4 Flux

23. 2.4.1 Fluid Example

24. 2.4.1 Fluid Example

25. - 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

26. - 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

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

28. 2.5.1 Simplifying Divergence

29. 2.5.1 Simplifying Divergence

30. 2.5.2 The Del Notation

31. - “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

32. - 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

33. 2.6 The Divergence Theorem

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

35. Chapter 3
Line Integrals
and the Curl

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

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

38. - Let t denote a unit vector that is tangent to the curve at P.
- Work done in three dimensions, if a force f(x, y, z) acts along the curve C.
3.2 Line Integrals Involving Vector Functions
^

39. 3.2 Line Integrals Involving Vector Functions

40. - t denote a unit vector that is tangent to the curve at P.
- Work done in three dimensions, if a force f(x, y, z) acts along the curve C.
3.2 Line Integrals Involving Vector Functions
^

41. - 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

42. 3.3.1 Line Integral around a Closed Path

43. - 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

44. 3.4 The Curl

45. 3.4 The Curl

46. 3.4 The Curl

47. 3.4.1 The Del Notation

48. - 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

49. 3.5 The Meaning of Curl

50. - 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

51. 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.

52. 3.6 Stokes’ Theorem

53. 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

54. 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.

55. Chapter 4
The Gradient

56. 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,^

57. 4.1 Line Integrals and the Gradient
- First, let us assume that the preceding relations hold.

58. 4.1 Line Integrals and the Gradient
- The result of line integral only depends on (x1
, y1
, z1
) and (x0
, y0
, z0
).
- Therefore, it is path independent.

59. 4.1 Line Integrals and the Gradient
- 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.

60. 4.1 Line Integrals and the Gradient
- 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.

61. 4.1 Line Integrals and the Gradient
- 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.

62. 4.1.1 The Del Notation

63. 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.

64. 4.2 Directional Derivatives and the Gradient

65. - 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.
4.2 Directional Derivatives and the Gradient
Δs

66. - 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

67. - 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.
4.3 Geometric Significance of the Gradient

68. - 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.
4.3 Geometric Significance of the Gradient

69. - 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

70. 4.4.1 The Laplacian Notation

71. 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.