Using the Divergence Theorem In Exercises 9-18, use the Divergence Theorem to evaluate
and find the outward flux of F through the surface of the solid S bounded by the graphs of the equations. Use a computer algebra system to verify your results.
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Multivariable Calculus
- Evaluate the circulation of G = xyi+zj+7yk around a square of side 9, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Prevs So F.dr-arrow_forwarda) Find the value(s) of (1+ i)2/3. b) Show that cos² z + sin? z = 1, for all z = x + iy, , y E R, V-I= i. c) Find a complex valued analytic function f(x, y) = u(x, y) + iv(x, y), whose real part u(x, y) = 213 – 3r?y – 6xy? + y°.arrow_forwardConsidering the scalar functions ∅ = ∅ (x, y, z) and ψ = ψ (x, y, z), find the following expression? ∇. (∇ ∅ × ∇ ψ) =?arrow_forward
- Evaluate the circulation of G = xyi + zj + 4yk around a square of side 4, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Jo F. dr =arrow_forwardVerify that u(x, y) = xy - x +y is harmonic in the complex plane and find a harmonic conjugate of u(x, y).arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forward
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- (5) Let ß be the vector-valued function 3u ß: (-2,2) × (0, 2π) → R³, B(U₁₂ v) = { 3u² 4 B (0,7), 0₁B (0,7), 0₂B (0,7) u cos(v) VI+ u², sin(v), (a) Sketch the image of ß (i.e. plot all values ß(u, v), for (u, v) in the domain of ß). (b) On the sketch in part (a), indicate (i) the path obtained by holding v = π/2 and varying u, and (ii) the path obtained by holding u = O and varying v. (c) Compute the following quantities: (d) Draw the following tangent vectors on your sketch in part (a): X₁ = 0₁B (0₂7) B(0)¹ X₂ = 0₂ß (0,7) p(0.4)* ' cos(v) √1+u² +arrow_forwardUse Stokes' Theorem to evaluate F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y2)i + (y + z2)j + (z + x2)k, C is the triangle with vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3).arrow_forwardUse Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above. (x + y?)i + (y + z?)j + (z + x2)k, C is the triangle with vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3). F(x, у, z)arrow_forward
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