Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUStotal derivative, the tangent to the curve r(λ) at any point is given bydrdλ = ∂r du∂u dλ + ∂r dv∂v dλ . (10.21)The two curves u =constantandv = constant passing through any point Pon S are called coordinate curves. Forthecurveu = constant, for example, wehave du/dλ = 0, and so from (10.21) its tangent vector is in the direction ∂r/∂v.Similarly, the tangent vector to the curve v = constant is in the direction ∂r/∂u.If the surface is smooth then at any point P on S the vectors ∂r/∂u and∂r/∂v are linearly independent and define the tangent plane T at the point P (seefigure 10.4). A vector normal to the surface at P is given byn = ∂r∂u × ∂r∂v . (10.22)In the neighbourhood of P , an infinitesimal vector displacement dr is writtendr = ∂r ∂rdu +∂u ∂v dv.The element of area at P , an infinitesimal parallelogram whose sides are thecoordinate curves, has magnitude∣ dS =∂r ∂r ∣∣∣∣ du ×∂u ∂v dv =∂r∣∂u × ∂r∂v ∣ du dv = |n| du dv. (10.23)Thus the total area of the surface is∫∫A =∂r∣∂u ∫∫∂v∣ |n| du dv, (10.24)Rwhere R is the region in the uv-plane corresponding to the range of parametervalues that define the surface.◮ Find the element of area on the surface of a sphere of radius a, and hence calculate thetotal surface area of the sphere.We can represent a point r on the surface of the sphere in terms of the two parameters θand φ:r(θ, φ) =a sin θ cos φ i + a sin θ sin φ j + a cos θ k,where θ and φ are the polar and azimuthal angles respectively. At any point P ,vectorstangent to the coordinate curves θ =constantandφ = constant are∂r= a cos θ cos φ i + a cos θ sin φ j − a sin θ k,∂θ∂r= −a sin θ sin φ i + a sin θ cos φ j.∂φ346R