Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

mathematical point of view, as we do below. In the following chapter, however, we

will discuss their geometrical definitions, which rely on the concept of integrating

vector quantities along lines and over surfaces.

Central to all these differential operations is the vector operator∇,whichis

calleddel(or sometimesnabla) and in Cartesian coordinates is defined by

∇≡i

∂

∂x

+j

∂

∂y

+k

∂

∂z

. (10.25)

The form of this operator in non-Cartesian coordinate systems is discussed in

sections 10.9 and 10.10.

10.7.1 Gradient of a scalar field

Thegradientof a scalar fieldφ(x, y, z) is defined by

gradφ=∇φ=i

∂φ

∂x

+j

∂φ

∂y

+k

∂φ

∂z

. (10.26)

Clearly,∇φis a vector field whosex-,y-andz- components are the first partial

derivatives ofφ(x, y, z) with respect tox,yandzrespectively. Also note that the

vector field∇φshould not be confused with the vector operatorφ∇, which has

components (φ ∂/∂x, φ ∂/∂y, φ ∂/∂z).

Find the gradient of the scalar fieldφ=xy^2 z^3.

From (10.26) the gradient ofφis given by

∇φ=y^2 z^3 i+2xy z^3 j+3xy^2 z^2 k.

The gradient of a scalar fieldφhas some interesting geometrical properties.

Let us first consider the problem ofcalculating the rate of change ofφin some

particular direction. For an infinitesimal vector displacementdr, forming its scalar

product with∇φwe obtain

∇φ·dr=

(

i

∂φ

∂x

+j

∂φ

∂y

+k

∂φ

∂z

)

·(idx+jdy+kdx),

=

∂φ

∂x

dx+

∂φ

∂y

dy+

∂φ

∂z

dz,

=dφ, (10.27)

which is the infinitesimal change inφin going from positionrtor+dr.In

particular, ifrdepends on some parameterusuch thatr(u) defines a space curve