Let F = rner, where n is any number, r = (x2 + y2 + z2) 112, and er = r - 1 (x, y, z) is the unit radial vector. (a) Calculate div(F). (b) Calculate the flux of F through the surf ace of a sphere of radius R centered at the origin. For which values of n is this flux independent of R? (c) Prove that V(rn) = n rn- ler. (d) Use (c) to show that Fis conservative for n =/=--1. Then show that F = r - 1 er is also conservative by computing the gradient of In r. (e) What is the value of fc F • ds, where C is a closed curve that does not pass through the origin? (f) Find the values of n for which the function <p = rn is harmonic.
Let F = rner, where n is any number, r = (x2 + y2 + z2) 112, and er = r - 1 (x, y, z) is the unit radial vector. (a) Calculate div(F). (b) Calculate the flux of F through the surf ace of a sphere of radius R centered at the origin. For which values of n is this flux independent of R? (c) Prove that V(rn) = n rn- ler. (d) Use (c) to show that Fis conservative for n =/=--1. Then show that F = r - 1 er is also conservative by computing the gradient of In r. (e) What is the value of fc F • ds, where C is a closed curve that does not pass through the origin? (f) Find the values of n for which the function <p = rn is harmonic.
Let F = rner, where n is any number, r = (x2 + y2 + z2) 112, and er = r - 1 (x, y, z) is the unit radial vector. (a) Calculate div(F). (b) Calculate the flux of F through the surf ace of a sphere of radius R centered at the origin. For which values of n is this flux independent of R? (c) Prove that V(rn) = n rn- ler. (d) Use (c) to show that Fis conservative for n =/=--1. Then show that F = r - 1 er is also conservative by computing the gradient of In r. (e) What is the value of fc F • ds, where C is a closed curve that does not pass through the origin? (f) Find the values of n for which the function <p = rn is harmonic.
Let F = rner, where n is any number, r = (x2 + y2 + z2) 112, and er = r - 1 (x, y, z) is the unit radial vector. (a) Calculate div(F). (b) Calculate the flux of F through the surf ace of a sphere of radius R centered at the origin. For which values of n is this flux independent of R?
(c) Prove that V(rn) = n rn- ler. (d) Use (c) to show that Fis conservative for n =/=--1. Then show that F = r - 1 er is also conservative by computing the gradient of In r. (e) What is the value of fc F • ds, where C is a closed curve that does not pass through the origin? (f) Find the values of n for which the function <p = rn is harmonic.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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