Electrodynamics/Vector Calculus Review

Electrodynamics

This page is going to review some of the necessary background information in physics and vector calculus. In this page, we will make extensive use of an analogy between vector fields and the flow of water so that you will gain intuitive understanding of the material.

The Del Operator

The del operator, ∇ is defined as follows:

\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:

\nabla \Phi ={\hat {x}}{\frac {\partial \Phi }{\partial x}}+{\hat {y}}{\frac {\partial \Phi }{\partial y}}+{\hat {z}}{\frac {\partial \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).

In addition, $\nabla \Phi \cdot {\hat {n}}$ gives the rate of change of $\Phi$ in the direction of ${\hat {n}}$ . Thus, the perpendicular lines to the gradient form an equipotential surface, or a surface where $\Phi$ are all equal.

Example 1.1
Plot and gradient of $F(x,y)=x^{2}-y^{2}$ The function $F(x,y)=x^{2}-y^{2}$ and $\nabla F$ are demonstrated in the associated diagram. The key concepts here are that the vectors of the gradient point towards the higher magnitude of $F(x,y)$ and that the vector represents the rate of change between the origin and head of the this vector. If $F(x,y)$ was a uniformly rigid surface and a perfect sphere was placed exactly at $F(2,0)$ it would move towards $F(-2,0)$ and eventually settle at $F(0,0)$ . In this case, if y is ever non-zero then the ball will eventually fall off the surface.

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:

\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. It measures how much the vector "diverges" from a single point. It measures the "sources" and "sinks" of the vector field. Imagine the velocity vector field of a pool of water. The faucets are places of high positive divergence, because it is the source of the water velocity field, and the sinks (drains) are places of high negative divergence, because this is where all the water is converging.

The Curl

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

\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. It measures the "curling" or the "rotation" of the vector field at a single point. Thus, going back to the pool analogy, a whirlpool would be a place with a large curl. In a steady flow, the curl is 0, since the field doesn't want to curl around that point.

The Laplacian

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

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

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

\nabla ^{2}A={\hat {x}}\nabla ^{2}A_{x}+{\hat {y}}\nabla ^{2}A_{y}+{\hat {z}}\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:

\nabla \cdot (\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. It turns out that every vector field that is divergence-free is the curl of another vector field.

The Curl of the Gradient

Likewise,

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

It also turns out that every vector field that has no curl is the gradient of a scalar field.

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:

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

Which we read as "The flux passing through dA, in the direction of n".

The concept of flux originally came from hydrodynamics. The flux passing through a small surface is the amount of liquid that flows through it. If the velocity field is big, then the flux would naturally turn out to be big. Also, if the surface is parallel to the velocity field, then the flux is 0, because no water is passing through the area. It turns out flux is a very useful concept in Electrodynamics also.

If we integrate this equation with respect to dA, we get the following:

\operatorname {Flux} (A,n)=\int _{A}v\cdot dA

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

\operatorname {NetFlux} =\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.

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

This formula is intuitively true because as we seen before, the divergence of a field is how much the field spreads from that point. If we add the spreading of every point inside a volume, that should be the amount that is leaving the volume through the closed surface. The flux through any closed surface of a divergence free field is 0.

Line Integration

Suppose we have a path $\gamma$ , composed of small length elements $ds$ . The line integral of a vector field A is how much the field lies along the path. Formally, it is given by:

\int _{\gamma }A\cdot ds

Suppose that A is a curl-free field. As seen above, this means it could be written as the gradient of a certain field. Say that $A=\nabla \Phi$ . It turns out that the line integral of A along any path connecting two points are the same. Moreover,

\int _{\gamma }A\cdot ds=\Phi (b)-\Phi (a)

where a and b are the start and end points of the path. Also, the line integral along a closed loop of such a field is always 0. A curl-free field is called conservative. The reason for this terminology came from mechanics: In mechanics, if you have a force field in space that is curl-free, you can always define a potential energy function, so that the work done in moving an object from a to b is the difference in potential energy. In this case, mechanical energy is conserved.

As we will see, the electrostatic field is a conservative field, whereas the magnetic field and the general electric field are not.

In general, however, the line integral of a vector field along a closed loop is nonzero. It turns out, however, that:

\int _{\gamma }F\cdot ds=\int _{A}(\nabla \times F)\cdot dA

. In other words, the line integral of a vector field is equal to the flux of the curl of the field through the loop. Through which surface are we finding the flux? Any surface! Any surface enclosed by the loop will suffice.

An intuitive understanding of this theorem can come as follows: the line integral of a closed loop is like how much the field wraps around the loop. However, the curl at a point gives how much the field rotates around it. Thus, the total integral over the surface gives the total "wrapping" around the loop.