Solution: In these exercises, F(x, y, z) denotes a vector field defined on asurface

Chapter 15, Problem 2

(choose chapter or problem)

In these exercises, \(F(x, y, z)\) denotes a vector field defined on a surface \(\sigma\) oriented by a unit normal vector field \(n(x, y, z)\), and \(\Phi\) denotes the flux of \(F\) across \(\sigma\).

(a) Assume that \(\sigma\) is parametrized by a vector-valued function \(r(u, v)\) whose domain is a region \(R\) in the uv-plane and that n is a positive multiple of

            \(\frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v}\)

Then the double integral over \(R\) whose value is \(\Phi\) is _______

(b) Suppose that \(\sigma\) is the parametric surface

\(r(u, v)=u i+v j+(u+v) k\left(0 \leq u^{2}+v^{2} \leq 1\right)\)

and that n is a positive multiple of

     \(\frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v}\)

Then the flux of   \(F(x, y, z) = xi + y j + zk\) across \(\sigma\) is \(\Phi\) = ________ .

Equation Transcription:

σ

Φ

(0 ≤ u2 + v2 ≤ 1)

Text Transcription:

sigma

F(x, y, z)

n(x, y, z)

Phi

partial r/partial u times partial r/partial v

r(u, v)

uv

r(u, v) = ui + v j + (u + v)k(0 <= u2 + v2 <= 1)