Electrodynamics/Vector Calculus Review

This page is going to review some of the necessary background information in physics and vector calculus.

The Del Operator

The del operator, ∇ is defined as follows:

$\mathrm{\nabla }=\hat{x}\frac{\partial }{\partial x}+\hat{y}\frac{\partial }{\partial y}+\hat{z}\frac{\partial }{\partial z}$

This operator, while confusing at first, is the method by which vectors and scalars can be differentiated.

The Gradient

When ∇ operates on a scalar field, like so:

$\mathrm{\nabla }\mathrm{\Phi }=\hat{x}\frac{\partial \mathrm{\Phi }}{\partial x}+\hat{y}\frac{\partial \mathrm{\Phi }}{\partial y}+\hat{z}\frac{\partial \mathrm{\Phi }}{\partial z}$

it simultaneously differentiates the scalar by all 3 axes (x, y, z). The result is called the "Gradient" of the scalar. The gradient is a vector that points in the direction in which the original scalar field is changing most rapidly (has the largest derivative).

The Divergence

The ∇ operator can be loosely treated as a "vector" whose components are the partial differential operators. If we operate on a vector field as a "dot product", we obtain:

$\mathrm{\nabla }\cdot A=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$

This is called the "Divergence" of the vector field.

The Curl

If we cross ∇ onto a vector field, we obtain another important operator:

$\mathrm{\nabla }\times A=\hat{x}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})+\hat{y}(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})+\hat{z}(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})$

The resulting vector is the "Curl" of the original vector.

The Laplacian

The gradient ∇Φ introduced above is a vector field. What happens if we take its divergence?

$\mathrm{\nabla }\cdot \mathrm{\nabla }\mathrm{\Phi }={\mathrm{\nabla }}^2\mathrm{\Phi }=\frac{{\partial }^2\mathrm{\Phi }}{\partial x^2}+\frac{{\partial }^2\mathrm{\Phi }}{\partial y^2}+\frac{{\partial }^2\mathrm{\Phi }}{\partial z^2}$

This important operator is known as the "Laplacian". The Laplacian is also defined for vector fields:

${\mathrm{\nabla }}^{2}A=\hat{x}{\mathrm{\nabla }}^2{A}_x+\hat{y}{\mathrm{\nabla }}^2{A}_y+\hat{z}{\mathrm{\nabla }}^2{A}_z$

The Divergence of the Curl

One might also expect to obtain an important operator by taking the divergence of a curl:

$\mathrm{\nabla }\cdot (\mathrm{\nabla }\times A)=\frac{\partial }{\partial x}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})+\frac{\partial }{\partial y}(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})+\frac{\partial }{\partial z}(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})=0$

While zero is certainly an important concept, it does not provide us with a useful operator. This identity, however, is interesting in its own right.

The Curl of the Gradient

Likewise,

$\mathrm{\nabla }\times \mathrm{\nabla }\mathrm{\Phi }=\hat{x}(\frac{{\partial }^2\mathrm{\Phi }}{\partial z\partial y}-\frac{{\partial }^2\mathrm{\Phi }}{\partial y\partial z})+\hat{y}(\frac{{\partial }^2\mathrm{\Phi }}{\partial z\partial x}-\frac{{\partial }^2\mathrm{\Phi }}{\partial x\partial z})+\hat{z}(\frac{{\partial }^2\mathrm{\Phi }}{\partial y\partial x}-\frac{{\partial }^2\mathrm{\Phi }}{\partial x\partial y})=0$

Vector Fields

Vector fields are 3 dimensional volumes, for which every point within that volume can be assigned a vector magnitude, based on some given rule. Gravity is one example of a vector field, where every point within a gravitational field is being pulled with some force magnitude towards the center. A vector field is denoted by a 3-dimensional function, such as A(x, y, z). The value of the function for each triplet is the magnitude of the vector field at that point.

Flux

In speaking of vector fields, we will discuss the notion of flux in general, and electric flux specifically. We can define the flux of a given vector field G(x, y, z), through an infinitesimal area dA, which has a normal vector n:

$\mathrm{Flux}(dA,n)=G\cdot ndA$

Which we read as "The flux passing through dA, in the direction of n". If we integrate this equation with respect to dA, we get the following:

$\mathrm{Flux}(A,n)=\underset{A}{\int }v\cdot dA$

We can also show (although the derivation can be long), that the flux traveling through into or out of a given vector field, G, can be given by the divergance of the vector field:

$\mathrm{NetFlux}=\mathrm{\nabla }\cdot G$

Let's say that we have an arbitrary volume, V, in a vector field, G, bounded by a surface, S, with surface-area, A. Gauss' Theorem states that the flux flowing into this volume is equal to the amount of flux flowing through the surface, S.

$\underset{V}{\int }(\mathrm{\nabla }\cdot G)dV=\underset{S,V}{\int }(v\cdot n)dA$

Divergence and Curl

We have shown that the divergence of an arbitrary vector A is given by:

$\mathrm{Divergence}(A)=\mathrm{\nabla }\cdot A$

and likewise, we define an operator called Curl that acts on a vector field and is defined as such:

$\mathrm{Curl}(A)=\mathrm{\nabla }\times A$

We will be using divergence and curl throughout the rest of the chapters on electromagnetism.