Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.11 EXERCISES 10.6 Prove that for a space curver=r(s), wheresis the arc length measured along the curve from a fixed point, th ...

VECTOR CALCULUS Use this formula to show that the area of the curved surfacex^2 +y^2 −z^2 =a^2 between the planesz=0andz=2ais πa ...

10.11 EXERCISES (a) For cylindrical polar coordinatesρ, φ, z, evaluate the derivatives of the three unit vectors with respect to ...

VECTOR CALCULUS 10.21 Paraboloidal coordinatesu, v, φare defined in terms of Cartesian coordinates by x=uvcosφ, y=uvsinφ, z=^12 ...

10.12 HINTS AND ANSWERS (a) Expresszand the perpendicular distanceρfromPto thez-axis in terms of u 1 ,u 2 ,u 3. (b) Evaluate∂x/∂ ...

VECTOR CALCULUS 10.23 The tangent vectors are as follows: foru= 0, the line joining (1, 0 ,0) and (− 1 , 0 ,0); forv= 0, the lin ...

11 Line, surface and volume integrals In the previous chapter we encountered continuously varying scalar and vector fields and d ...

LINE, SURFACE AND VOLUME INTEGRALS Each of the line integrals in (11.1) is evaluated over some curveCthat may be either open (Aa ...

11.1 LINE INTEGRALS A similar procedure may be followed for the third type of line integral in (11.1), which involves a cross pr ...

LINE, SURFACE AND VOLUME INTEGRALS y x (i) (ii) (1,1) (iii) (4,2) Figure 11.1 Different possible paths between the points (1, 1) ...

11.1 LINE INTEGRALS
Evaluate the line integralI= ∮ Cxdy,whereCis the circle in thexy-plane defined by x^2 +y^2 =a^2 ,z=0. Adopt ...

LINE, SURFACE AND VOLUME INTEGRALS integral in (11.1). If a loop of wireCcarrying a currentIis placed in a magnetic fieldBthen t ...

11.2 CONNECTIVITY OF REGIONS (a) (b) (c) Figure 11.2 (a) A simply connected region; (b) a doubly connected region; (c) a triply ...

LINE, SURFACE AND VOLUME INTEGRALS y d c a b x S R T C U V Figure 11.3 A simply connected regionRbounded by the curveC. These id ...

11.3 GREEN’S THEOREM IN A PLANE y=y 2 (x) be the equations of the curvesSTUandSVUrespectively. We then write ∫∫ R ∂P ∂y dx dy= ∫ ...

LINE, SURFACE AND VOLUME INTEGRALS y x C 1 C 2 R Figure 11.4 A doubly connected regionRbounded by the curvesC 1 andC 2. to the r ...

11.4 CONSERVATIVE FIELDS AND POTENTIALS
Evaluate the line integral I= ∮ C [(exy+cosxsiny)dx+(ex+sinxcosy)dy], around the ellips ...

LINE, SURFACE AND VOLUME INTEGRALS which shows that we requirea·drto be an exact differential: condition (iv). From (10.27) we c ...

11.5 SURFACE INTEGRALS independent of the path taken. Sinceais conservative, we can writea=∇φ. Therefore,φ must satisfy ∂φ ∂x =x ...

LINE, SURFACE AND VOLUME INTEGRALS S S V C dS dS (a)(b) Figure 11.5 (a) A closed surface and (b) an open surface. In each case a ...