Gradient divergence and curl and expansion formulae involving them
Book 📚 pdf
Notes
Gradient, Divergence and Curl:
the Basics We first consider the position vector,
r: r = x x + y y + z z ,
where x, y, and z are rectangular unit vectors. Since the unit vectors for rectangular coordinates are constants,
we have for
dr: dr = dx x + dy y + dz z .
The operator, del: Ñ, is defined to be (in rectangular coordinates):
Ñ( ) = x ¶( ) /¶x + y ¶( ) /¶y + z ¶( ) /¶z .
Note: the unit vectors are placed here in front of the operation, e.g., ¶( ) /¶x , to show that the operation is not performed on the unit vector itself. After the operations, the unit vectors will be placed after as is more usual. This operator operates as a vector.
1. Gradient If the del operator, Ñ, operates on a scalar function, f(x,y,z), we get the gradient:
Ñf = (¶f/¶x) x + (¶f/¶y) y + (¶f/¶z) z .
We can interpret this gradient as a vector with the magnitude and direction of the maximum change of the function in space. We can relate the gradient to the differential change in the function:
df = (¶f/¶x) dx + (¶f/¶y) dy + (¶f/¶z) dz = Ñf · dr = df .
Note: We will use this relation to determine the del operator, Ñ, in other coordinate systems later.
---------------------------
Since the del operator should be treated as a vector, there are two ways for a vector to multiply another vector: dot product and cross product. We first consider the dot product:
2. Divergence The divergence of a vector is defined to be:
Ñ · A = [x ¶ /¶x + y ¶ /¶y + z ¶ /¶z ] · [Ax x + Ay y + Az z] = (¶Ax /¶x) + (¶Ay /¶y) + (¶Az /¶z) ;
since the rectangular unit vectors are constant, ¶x/¶x = 0 (etc.). This will not necessarily be true for unit vectors in other coordinate systems. We'll see examples of this soon. To get some idea of what the divergence of a vector is, we consider Gauss' theorem (sometimes called the divergence theorem).
We start with:
òòò Ñ · A dV = òòò [(¶Ax /¶x) + (¶Ay /¶y) + (¶Az /¶z)] dx dy dz = òòò [(¶Ax /¶x)dx dydz + (¶Ay /¶y)dy dxdz + (¶Az /¶z)dz dxdy]
. We can see that each term as written in the last expression gives the value of the change in vector A that cuts perpendicular through the surface. For instance, consider the first term:
(¶Ax/¶x)dx dydz .
The first part: (¶Ax/¶x)dx gives the change in the x-component of A and the second part, dydz, gives the yz surface (or x component of the surface, Sx) where we define the direction of the surface vector as that direction that is perpendicular to its surface. The other two terms give the change in the component of A that is perpendicular to the xz (Sy) and xy (Sz) surfaces.
We thus can write: òòò Ñ · A dV = òò closed surface A· dS where the vector S is the surface area vector. Thus we see that the volume integral of the divergence of vector A is equal to the net amount of A that cuts through (or diverges from) the closed surface that surrounds the volume over which the volume integral is taken. Hence the name divergence for Ñ · A .