Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

total derivative, the tangent to the curver(λ) at any point is given by

dr

dλ

=

∂r

∂u

du

dλ

+

∂r

∂v

dv

dλ

. (10.21)

The two curvesu=constantandv= constant passing through any pointP

onSare calledcoordinate curves.Forthecurveu= constant, for example, we

havedu/dλ= 0, and so from (10.21) its tangent vector is in the direction∂r/∂v.

Similarly, the tangent vector to the curvev= constant is in the direction∂r/∂u.

If the surface is smooth then at any pointP onSthe vectors∂r/∂uand

∂r/∂vare linearly independent and define thetangent planeTat the pointP(see

figure 10.4). A vector normal to the surface atPis given by

n=

∂r

∂u

×

∂r

∂v

. (10.22)

In the neighbourhood ofP, an infinitesimal vector displacementdris written

dr=

∂r

∂u

du+

∂r

∂v

dv.

Theelement of areaatP, an infinitesimal parallelogram whose sides are the

coordinate curves, has magnitude

dS=

∣

∣

∣

∣

∂r

∂u

du×

∂r

∂v

dv

∣

∣

∣

∣=

∣

∣

∣

∣

∂r

∂u

×

∂r

∂v

∣

∣

∣

∣du dv=|n|du dv. (10.23)

Thus the total area of the surface is

A=

∫∫

R

∣

∣

∣

∣

∂r

∂u

×

∂r

∂v

∣

∣

∣

∣du dv=

∫∫

R

|n|du dv, (10.24)

whereRis the region in theuv-plane corresponding to the range of parameter

values that define the surface.

Find the element of area on the surface of a sphere of radiusa, and hence calculate the

total surface area of the sphere.

We can represent a pointron the surface of the sphere in terms of the two parametersθ
andφ:

r(θ, φ)=asinθcosφi+asinθsinφj+acosθk,

whereθandφare the polar and azimuthal angles respectively. At any pointP,vectors
tangent to the coordinate curvesθ=constantandφ= constant are

∂r

∂θ

=acosθcosφi+acosθsinφj−asinθk,

∂r

∂φ

=−asinθsinφi+asinθcosφj.