Prof. Alex Small Phys 234 Vector Calculus and Maxwell's
[email protected] This document starts with a summary of useful facts from vector calculus, and then uses them to derive Maxwell's equations. First, definitions of vector operators. 1. Gradient Operator: The gradient operator is something that acts on a function f and produces a vector whose components are equal to derivatives of the function. It is defined by: @f @f @f rf = ^x + ^y + ^z (1) @x @y @z 2. Divergence: We can apply the gradient operator to a vector field to get a scalar function, by taking the dot product of the gradient operator and the vector function. @V @V @V r ⋅ V~ (x; y; z) = x + y + z (2) @x @y @z 3. Curl: We can take the cross product of the gradient operator with a vector field, and get another vector that involves derivatives of the vector field. @V @V @V @V @V @V r × V~ (x; y; z) = ^x z − y + ^y x − z + ^z y − x (3) @y @z @z @x @x @y 4. We can also define a second derivative of a vector function, called the Laplacian. The Laplacian can be applied to any function of x, y, and z, whether the function is a vector or a scalar. The Laplacian is defined as: @2f @2f @2f r2f(x; y; z) = r ⋅ rf = + + (4) @x2 @y2 @z2 Notes that the Laplacian of a scalar function is equal to the divergence of the gradient. Second, some useful facts about vector fields: 1. If a vector field V~ (x; y; z) has zero curl, then it can be written as the gradient of a scalar function f(x; y; z).