BASIC ELEMENTS OF VECTOR CALCULUS

Definitions

In the following, S is a scalar function of (x,y,z), S(x,y,z),
and V and W are vector functions of (x,y,z):
del ∇ is gradient, a vector in the chosen coordinate system
divergence is  ∇•V  a scalar
curl is ∇xV  a vector

V = V_x(x,y,z)x + V_y(x,y,z)y + V_z(x,y,z)z,

where x, y, and z are unit vectors in the x, y, and z directions respectively [rectangular], and r, Θ, and Φ are unit vectors in the directions of increasing r, Θ, and Φ [polar and cylindrical, and spherical coordinate systems].

The Del Operator

Let ∇ = del operator. In Cartesian form it is:

∇ = (∂[]/∂x)x + (∂[]/∂y)y + (∂[]/∂z)z,

where ∂[]/∂x_i indicates the operation of taking the Partial Derivative with respect to the i'th Cartesian position variable.

Gradient

Let the gradient be defined as: ∇S =(∂S/∂x)x + (∂S/∂y)y + (∂S/∂z)z = gradS.

The gradient gives the change in S in the direction of that greatest change, and hence is a vector.  The significance of the gradient is best seen in the Gradient Theorem.  It is the multivariable analog of the Fundamental Theorem of Calculus.  It also allows us to find a conservative force if we know the potential energy associated with it.  To see this, consider the change in the potential energy due to an infinitesimal position displacement, dx, in the x-direction of a Cartesian coordinate system.

U(x + dx, y,z) = U(x,y,z) + dU =  U(x,y,z) - dW = U(x,y,z) - F_xdx,

where we have used dW = F_xdx = - dU.  Rearranging terms, we have

[U(x + dx, y,z) - U(x,y,z)]/dx = -F_x.

The left hand side of this equation, in the limit dx --> 0, is just the Partial Derivative of U with respect to x, so we have

F_x = -∂U/∂x,

and similarly for F_y and F_z.  Thus, taking all three components,

F = -∇U = -gradU.

A conservative force is the negative of the gradient of the potential energy associated with it.  Since the gradient of a function gives the change in the function in the direction of the greatest change, the force is in the direction of greatest negative change in the potential energy.  Thus, from the point of view of energy, a particle acted on by the force is accelerated in the direction which maximizes its decrease in potential energy.

Divergence

Let the divergence be defined as: ∇•V = (∂V_x/∂x)+(∂V_y/∂y)+ (∂V_z/∂z)= divV (a scalar).  The significance of the divergence is best seen in the Divergence Theorem (or Gauss' Theorem).

Curl

Let the curl be defined as: ∇xV = (∂V_z/∂y - ∂V_y/∂z)x + (∂V_x/∂z - ∂V_z/∂x)y + (∂V_y/∂x - ∂V_x/∂y) z = curlV.
The significance of the curl is best seen in the Curl Theorem (sometimes incorrectly called Stokes' Theorem).

Laplacian

Let the Laplacian be defined as: ∇²S = ∇•(∇S) = (∂²S/∂x² + ∂²S/∂y² + ∂²S/∂z²)

Biharmonic operator

Let the Biharmonic operator be defined as: ∇⁴S = (∂⁴S/∂x⁴ + ∂⁴S/∂y⁴ + ∂⁴S/∂z⁴)

 Stokes' Theorem

Stokes' Theorem is a very general integral theorem, which contains the divergence, gradient, and curl theorems as special cases.

Identities involving the Del Operator

∇•(V+W) = ∇•V + ∇•W

∇•(SV) = (&nablaS)•V + S(∇•V)

∇• (V+W) = (∇•V) + (∇•W)

∇•(SV) = (∇S)•V + S(∇•V)

∇• (V x W) = W• (∇ x V) - V• (∇ x W)

∇ x (V x W) = (W•∇)V - (V• ∇ )W + (∇•W)V - (∇•V)W

∇ (V•W) = (W•∇ )V + (V•∇)W + W x (∇ x V) + V x (∇ x W)

∇ x (∇ x V) = ∇(∇•V) - ∇²V

∇ x (∇S) = 0

∇• (∇ x V) = 0

The Del Operator in Spherical Coordinates

∇ = (∂/∂r)r + (1/r)(∂/∂Θ)Θ + (1/[r sinΦ])(∂/∂Φ)Φ

∇²S = (1/r²)(∂/∂r)(r² ∂S/∂r) + (1/[r²sin &Theta])(∂/∂Θ)(sin Θ ∂S/∂Θ) + (1/[r²sin² Θ])(∂²S/∂Φ²)

You will find expressions for spherical coordinates div and curl here.

The Del Operator in Cylindrical Coordinates

∇ = (∂/∂)r + (1/r)(∂/∂Θ)Θ + (∂/∂z)z

∇²S = (1/r)(∂/∂r)(r ∂S/∂r) + (1/r²) (∂²S/∂Θ²) + (∂²S/∂z²)

You will find expressions for cylindrical coordinates div and curl here.

Vector fields in cylindrical and spherical coordinates here.

div and curl in spherical coordinates here.