Special case of surface integrals of scalar-valued

Chapter 11, Problem 71AE

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QUESTION:

Suppose that a surface S is defined as \(z=g(x, y)\) on a region R. Show that \(\mathbf{t}_{x} \times \mathbf{t}_{y}=\left(-z_{x},-z_{y}, 1\right)\) and that \(\iint_{S} f(x, y, z) d S=\) \(\iint_{R} f(x, y, z) \sqrt{z_{x}^{2}+z_{y}^{2}+1} d A\).

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QUESTION:

Suppose that a surface S is defined as \(z=g(x, y)\) on a region R. Show that \(\mathbf{t}_{x} \times \mathbf{t}_{y}=\left(-z_{x},-z_{y}, 1\right)\) and that \(\iint_{S} f(x, y, z) d S=\) \(\iint_{R} f(x, y, z) \sqrt{z_{x}^{2}+z_{y}^{2}+1} d A\).

ANSWER:

Solution 71AEStep 1

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