7. (1 point) Show that \(\Phi(u, v) = (7u+4, u-v, 13u+v)\) parametrizes the plane \(2x - y - z = 8\). Then: (a) Calculate \(T_u\), \(T_v\), and \(n(u, v)\). (b) Find the area of \(S = \Phi(D)\), where \(D = \{(u, v) : 0 \le u \le 7, 0 \le v \le 7\}\). (c) Express \(f(x, y, z) = yz\) in terms of \(u\) and \(v\) and evaluate \(\iint_S f(x, y, z) \,dS\).
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We have: uv + 7u + 4u - v + 13u + v = uv + 24u Now, let's express x, y, and z in terms of u and v. From the equation of the plane 2x - y - z = 8, we can solve for x, y, and z: 2x = y + z + 8 x = (y + z + 8)/2 So, we have: x = (y + z + 8)/2 y = y z = z Now, Show more…
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