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Stoke's theorem says: I (3 x 2). a3 = fh.at Verify Stoke's theorem for the vector...
8.) (16 pts.) Verify Stoke's Theorem for the Vector Field F (, y, z) = (-y)i + (-2)5+(z)k, where Surface S is that portion of the paraboloid z= 6 – 12 – yº, which lies above the plane z = 2.
Problem 3. Vector Calculus (30 points) Set up the equations for Stoke's theorem for the vector field B frcosd + psinp in the surface shown below. No need to evaluate the equations (a) Write the closed contour integral equation and simplify any vector algebra (gradient, divergence, curl, dot product, etc.). Do not evaluate the integral. (15 points) (b) Write the open surface integral equation and simplify any vector algebra (gradient, divergence, curl, dot product, etc.). Do not evaluate the integral....
10.) (19 pts.) Verify Stoke's Theorem for the Vector Field F(x, y, z) = (-y)ī+(x)]+(z)k, where Surface S is that portion of the paraboloid z = 6 - 22 - y2, which lies above the plane z = 2.
a) Complete the statement of: Stoke's Theorem: Let S be an oriented surface bounded by a piecewise smooth simple closed curve with a positive orientation (i.e. clockwise relative to N). If F(x, y, z)=(M(x,y,z), N(x, y, z), P(x, y, z)) where M, N, and P have continuous partial derivatives in an open region containing Sand C, then: b) Use Stoke's theorem to write as an iterated integral, J. (y, -2', 1)odr where is the circle of radius 1 in the...
Q4. 8pnts]If you haven't explored it yet, here is a magical property of the Stoke's theorem Suppose we have a vector field F(x,y, z) = -yi+ xj+ zk. Also, let C: x2y2 R2 for some R 0 be the curve in the xy-plane. Now, verify the Stoke's theorem when: (a) The surface S is given by the upper hemisphere 2y z2= R2,z0. R2 - y2, z 2 0. (b) The surface S is given by the paraboloid (c) The surface...
4. Stoke's Theorem: Consider a vector field F = (1,1)+(1,0) + (0,0). tyle +rin the unit square bordered by (0,0) + (0,1) ► (a) What is the curl of the vector field F? [2 points) (b) What is the path integral of the vector field around the unit square? [5 points) (c) Show your answers to the previous parts are consistent with Stoke's Theorem. HINT: consider the right-hand rule. (3 points]
1. Gauß theorem / Divergence Theorenm Given the surface S(V) with surface normal vector i of the volume V. Then, we define the surface integral fs(v) F , df = fs(v) F-ndf over a vector field F. S(V) a) Evaluate the surface integral for the vector field ()ze, - yez+yz es over a cube bounded by x = 0,x = 1, y = 0, y = 1, z 0, z = 1 . Then use Gauß theorem and verify it....
4. (18 points) Verify Stokes' Theorem in finding the counterclockwise circulation of the vector field, F - (r-i + (42)j + (r) k around the curve, C, where C is the triangular path determined by the points (6,0,0),(0,-4,0),and (0,0,10) . (i.e. calculate the circulation % F.iF directly, and then by using Stokes' Theorem and doing a surface integral.) Which way was easier? (Hint: You will need to find the equation of the plane that goes through these three points.) 4....
Problem (10 marks) Verify the Divergence Theorem for the vector field F(x, y, z) = (y,1,-) on the region E bounded by the planes y + : = 2 := 0 and the cylinder r +y = 1. Surface Integral: 6 marks) Triple Integral: (4 marks)
Verify that the line integral and the surface integral of Stokes Theorem are equal far the following vector field, surface S, and closed curve C. Assume that C has counterlockwise orientation and S has a consistentorientation F = 〈y,-x, 11), s is the upper half of the sphere x2 + y2 +22-1 and C is the circle x2 + y2-1 in the xy-plane Construct the line integral of Stokes' Theorem using the parameterization r(t)= 〈cost, sint, O. for 0 sts2r...