Chapter 1. Vector Analysis 1.3 Integral Calculus 1.3.1 Line, Surface, and Volume Integrals (a) Line Integrals. A line integral is an expression of the form If the path P in question forms a closed loop (that is, if b = a), Example 1.6 (path 1) (path 2) For the loop that goes out (1) and back (2) Line, Surface, and Volume Integrals (b) Surface Integrals. A surface integral is an expression of the form perpendicular to the surface If the surface is closed Example 1.7 Line, Surface, and Volume Integrals (c) Volume Integrals. A volume integral is an expression of the form A vector functions Example 1.8 1.3.2 The Fundamental Theorem of Calculus OR Fundamental theorem the integral of a derivative over an interval is given by the boundaries (the value of the function at the end points) In vector calculus, there are three derivatives (gradient, divergence, and curl), Fundamental theorem for gradients Fundamental theorem for divergences Gauss's theorem, Green's theorem, or, simply, the divergence theorem Fundamental theorem for curls Stokes’ theorem 1.3.3 The Fundamental Theorem for Gradients The integral (here a line integral) of a derivative (here the gradient) is given by the value of the function at the boundaries (a and b). Example 1.9 Check the fundamental theorem for gradients. Let's go out along the x axis, step (i), and then up, step (ii) Are they consistent with the fundamental theorem? Yes! T(b) – T(a) = 2 – 0 = 2 Now, calculate the same integral along path (iii): 1.3.4 The Fundamental Theorem for Divergences The integral (here a volume integral) of a derivative (here the divergence) is given by the value of the function at the boundaries (surface). Gauss's theorem Green's theorem Divergence theorem If v represents the flow of an incompressible fluid, then the flux of v is the total amount of fluid passing out through the surface, per unit time. Example 1.10 Check the divergence theorem using the function In this case, To evaluate the surface integral, ~ 1.3.5 The Fundamental Theorem for Curls The integral (here a surface integral) of a derivative (here the curl) is given by the value of the function at the boundaries (perimeter of the surface). Stokes’ theorem For dl, which way are we supposed to go around (clockwise or counterclockwise)? For da, which way does it point? which way is "out?" Consistency in Stokes' theorem Let’s Keep the right-hand rule Example 1.11 1.3.6 Integration by Parts Integrating both sides We can transfer the derivative from g to f, at the cost of a minus sign and a boundary term. Example 1.12 Note: by the product rules of vector calculus From the divergence theorem Same as the integration by parts 1.4 Curvilinear Coordinates A.1 (orthogonal) Curvilinear Coordinates: (,,uvw ) A.2 Notation 1.4.1 Spherical Polar Coordinates (,r , ) polar angle xr sin cos yr sin sin zr cos azimuthal angle Beware that The unit vectors, , at a particular point P, change direction as P moves around. Do not naively combine the spherical components of vectors associated with different points The unit vectors themselves are functions of position Do not take outside the integral Spherical Polar Coordinates (,r , ) Infinitesimal displacement Infinitesimal volume Infinitesimal surfaces depend on the orientation of the surface r is constant is constant Example 1.13 Find the volume of a sphere of radius R. Vector derivatives in (, r , ) Coordinates 1.4.2 Cylindrical Coordinates (,s ,z ) xs cos ys sin zz azimuthal angle Appendix A: Vector Calculus in Curvilinear Coordinates fg h x, y, z 1 1 1 s, , z 1r 1 r, , 1rr sin.