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10. Let F be the vector field F(x, y, z) = r/|r|5, where r = xi+yj + zk. If S is the sphere x² + y² +2²=16, evaluate ff, F-dS by inspection. π/4

10. Let F be the vector field F(x, y, z) = r/|r|5, where r = xi+yj + zk. If S is the sphere x² + y² +2²=16, evaluate ff, F-dS by inspection. π/4

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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10. Let F be the vector field F(x, y, z) = r/r5, where r = xi+yj + zk. If S is the sphere x² + y² + z² = 16, evaluate ff, F. dS by inspection. ㅠ/4 11. Let C be the triangle in the ry plane with vertices at (0,0), (1,0), and (0,1), oriented counter- clockwise. (a) Evaluate f2y² dx + 2x dy directly as a line integral. 1/3 (b) Evaluate the line integral in (a) by using Green's Theorem and then evaluating a double integral. 12. Show that the vector field F (3x²y²z+1,2x³yz + 2, x³y² + 3) has zero curl. Then find a x³y²z+x+2y+3z = potential V with F = VV. 13. Evaluate f(y + e) dx - xdy, where C is the square with vertices (0,0), (1,0), (1, 1), and (0,1), oriented counterclockwise. -2 14. Let W be the solid region 3 ≤ z ≤4-(x² + y²), S its boundary surface, and F the vector field F = (y + x,y-x, 1-32). Verify the divergence theorem ffw div F dV=ffs F. dS for F and W by calculating both sides and showing that they are equal. -π/2=-π/2 15. Let v(x, y, z) be the vector field v = x²i+ 2xyj. Evaluate the flux integral foven ds, with C the circle (x-1)² + y² = 1 and en, the outward normal to this curve. 16. Let a(x, y, z) be the vector field a = (x+y)i + (y-x)j + zk. (a) Evaluate fa. dr for C the straight line segment joining (0,1,2) to (-1,2,0). (b) Evaluate fa dr for C the circle x² + y² = 4, oriented clockwise, in the plane z = 0. = 4, z ≤ 0. Using (b), check that Stokes' Theorem (c) Let S be the hemisphere x² + y² + 2² Sa dr=ffs curla. dS is satisfied here. 8п=8п 17. Suppose that f(x, y, z) and F(x, y, z) are respectively a function and a vector field, each with two continuous derivatives. Show that curl(ƒ) = 0 and that div(curl(F)) = 0. JJ₁₁ curl F. ds = ₁ -1 18. Let S₁ be the disk x² + y² ≤ 1 in the plane z = 0, let S₂ be the hemisphere x² + y² + z² = 1, z ≥ 0, and let F be a twice differentiable vector field defined in all of space. (a) Show that Stokes' Theorem implies that curl F. ds, 8п if both surfaces are oriented with upward pointing normal vectors. (b) Derive the equality (**) from the Divergence Theorem. Hint: use the previous problem. (**)
Transcribed Image Text:10. Let F be the vector field F(x, y, z) = r/r5, where r = xi+yj + zk. If S is the sphere x² + y² + z² = 16, evaluate ff, F. dS by inspection. ㅠ/4 11. Let C be the triangle in the ry plane with vertices at (0,0), (1,0), and (0,1), oriented counter- clockwise. (a) Evaluate f2y² dx + 2x dy directly as a line integral. 1/3 (b) Evaluate the line integral in (a) by using Green's Theorem and then evaluating a double integral. 12. Show that the vector field F (3x²y²z+1,2x³yz + 2, x³y² + 3) has zero curl. Then find a x³y²z+x+2y+3z = potential V with F = VV. 13. Evaluate f(y + e) dx - xdy, where C is the square with vertices (0,0), (1,0), (1, 1), and (0,1), oriented counterclockwise. -2 14. Let W be the solid region 3 ≤ z ≤4-(x² + y²), S its boundary surface, and F the vector field F = (y + x,y-x, 1-32). Verify the divergence theorem ffw div F dV=ffs F. dS for F and W by calculating both sides and showing that they are equal. -π/2=-π/2 15. Let v(x, y, z) be the vector field v = x²i+ 2xyj. Evaluate the flux integral foven ds, with C the circle (x-1)² + y² = 1 and en, the outward normal to this curve. 16. Let a(x, y, z) be the vector field a = (x+y)i + (y-x)j + zk. (a) Evaluate fa. dr for C the straight line segment joining (0,1,2) to (-1,2,0). (b) Evaluate fa dr for C the circle x² + y² = 4, oriented clockwise, in the plane z = 0. = 4, z ≤ 0. Using (b), check that Stokes' Theorem (c) Let S be the hemisphere x² + y² + 2² Sa dr=ffs curla. dS is satisfied here. 8п=8п 17. Suppose that f(x, y, z) and F(x, y, z) are respectively a function and a vector field, each with two continuous derivatives. Show that curl(ƒ) = 0 and that div(curl(F)) = 0. JJ₁₁ curl F. ds = ₁ -1 18. Let S₁ be the disk x² + y² ≤ 1 in the plane z = 0, let S₂ be the hemisphere x² + y² + z² = 1, z ≥ 0, and let F be a twice differentiable vector field defined in all of space. (a) Show that Stokes' Theorem implies that curl F. ds, 8п if both surfaces are oriented with upward pointing normal vectors. (b) Derive the equality (**) from the Divergence Theorem. Hint: use the previous problem. (**)
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