Vectors Calaulus 1.35

Practice 3 Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work. Problem 1    Please answer the following true or false.  If false, explain why or provide a counter example.  If true explain why.  A.  Let f(x,y,z) be a function with continuous second order partial derivatives and let F(x,y,z) be the gradient of f(x,y,z).  If S is the ellipsoid              x2                      +  y2 + z2  =  1             4 oriented outward, then          Solution False Using the divergence theorem, we have         If gradf(x,y,z)  =  F, then          divF  =  fxx + fyy + fzz  There theorem would be true if the function was harmonic, however if it not harmonic.  All bets are off.  For example, if         f(x,y,z)  =  1/6 (x2 + y2 + z2) and          divF  =  1/6 (2 + 2 + 2)  =  1 Hence the integral represents the volume of the ellipse which is certainly not zero.   B.  Let F(x,y) be a conservative vector field, then           Solution   True,   The first line integral traces out the line segment from (0,1) to (1,0) and the second traces out the quarter-circle from (0,1) to (1,0).  Notice that in the first integral         r1(t)  =  (1 - t)i + tj         r1'(t)  =  -i + j   and in the second integral          r2(q)  =  (cos q)i + (sin q)j         r1'(t)  =  (-sin q)i +(cos q) j   By the fundamental theorem of line integral, the integral is independence of path, hence the two integrals are equal. Problem 2 Show that  for any closed surface S.   Solution We use Stokes' Theorem.  We have         Where C is the boundary of the surface S.  But since S is a closed surface, it has not boundary.  Hence C is a curve of zero length and the right hand integral is zero. Problem 3   A fish starting at the origin swims in a straight path to the point (0.2,0.1,0.3), then changes direction and swims along the circular path through the point (0.5,0.2,0.6) and the point (0.7,0.5,0.8), and finally changes directions heading straight to the point (1.5,1,2).  The current can by represented by the vector field         F(x,y,z)  =  (2x + 2z)i + (1 - 3z)j + (2x - 3y + 5)k Find the total work done by the current. Solution The important thing to not here is that         Since F is conservative, we can use the fundamental theorem of line integrals.  We seek a potential function f.  We have         fx  =  2x + 2z Integrating with respect to x gives         f  =  x2 + 2xz + C(y,z) Now taking the derivative with respect to y gives         fy  =  Cy(y,z)  =  1 - 3z Integrating with respect to y gives         C(y,z)  =  y - 3yz + C(z) so that         f  =  x2 + 2xz + y - 3yz + C(z) Now we take the derivative with respect to z to get         2z - 3y + C'(z)  =  2x - 3y + 5 so that          C'(z)  =  5 Integrate with respect to z to get         C(z)  =  5z Hence          f(x,y,z)  =   x2 + 2xz + y - 3yz + 5z The fundamental theorem of line integrals gives that the integral is         f(1.5,1,2) - f(0.2,0.1,0.3)  =           [(1.5)2 + 2(1.5)(2) + 1 - 3(1)(2) + 5(2)] - [(0.2)2 + 2(0.2)(0.3) + 0.1 - 3(0.1)(0.3) + 5(0.3)]         =  11.58 Problem 4 Evaluate where F is the vector field          and S is the rectangular solid with vertices (0,0,0), (1,0,0), (1,2,0), (0,2,0), (0,0,3), (1,0,3), (1,2,3), (0,2,3). Solution We use the divergence theorem.  We have         divF  =  1 - 1 + 2  =  2 We have         Since the integrand of the right hand side is just a constant, its value is equal to the constant times the volume of the solid.  Since the solid is a rectangular solid with side lengths 1, 2, and 3, we have         2(Volume E)  =  2(1)(2)(3)  =  12 Problem 5 Find the work done by sailing a ship from the point (2,3) the the point (-1,2) against the wind with velocity field         F(x,y)  =  yi + (3x + 2y)j Solution First notice that F is not a conservative vector field.  We need to parameterize the curve and perform the line integral.  The curve can be parameterized by         r(t)  =  (2 + (-1 - 2)t)i + (3 + (2 - 3))j  =  (2 - 3t)i + (3 - t)j We have         dr  =  -3i - j                 F(t)  =  (3 - t)i + (3(2 - 3t) + 2(3 - t))j  =  (3 - t)i + (12 - 11t)j  The integrand becomes         F . dr  =  -3(3 - t) + (-1)(12 - 11t)  =  -21 + 14t Now we integrate         Problem 6 Find the flux of F through the surface S where          F(x,y,z)  =  3zi - 4j + yk and S is the part of the plane         x + y + z  =  1 in the first octant with upwardly pointing unit normal. Solution We use Stokes' theorem.  We have         We can write the surface as         z  =  1 - x - y Using Stokes theorem we get