2. (a) Let r = (x, Y, z) and r = ||r||. Assuming r cannot equal 0, is there a value of p for which the vector field f(r)=r/r^P is solenoidal? You must fully justify your answer. (b) Show that if f and g are twice differentiable scalar fields then V²(f g) = ƒ V^2g+ g V²f + 2Vƒ · Vg .

2. (a) Let r = (x, Y, z) and r = ||r||. Assuming r cannot equal 0, is there a value of p for which the vector field f(r)=r/r^P is solenoidal? You must fully justify your answer. (b) Show that if f and g are twice differentiable scalar fields then V²(f g) = ƒ V^2g+ g V²f + 2Vƒ · Vg .

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Vectors In Two And Three Dimensions

Section9.FOM: Focus On Modeling: Vectors Fields

Problem 11P

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Question

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2. (a) Let r = (x, Y, z) and r = ||r||. Assuming r cannot equal 0, is there a value of p for which the vector field f(r)=r/r^P is solenoidal? You must fully justify your answer.

(b) Show that if f and g are twice differentiable scalar fields then V²(f g) = ƒ V^2g+ g V²f + 2Vƒ · Vg .

3. (a) Show that any vector field of the form h(x, Y, z) = f(x) i+g(y)j+h(z) k, where f, g, h are differentiable functions, is irrotational.

(b) Determine whether there is a vector field g such that V x g = xi+yj+zk.

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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