6. Vector Integral Calculus in Space 6A. Vector Fields in Space xi +yj +zk 6A-1 Describe geometrically the following vector fields: a) b) -xi-zk P 6A-2 Write down the vector field where each vector runs from (x, y, z) to a point half-way towards the origin. 6A-3 Write down the velocity field F representing a rotation about the x-axis in the direction given by the right-hand rule (thumb pointing in positive x-direction), and having constant angular velocity w. 6A-4 Write down the most general vector field all of whose vectors are parallel to the plane 3x -4y + z = 2. 6B. Surface Integrals and Flux 6B-1 Without calculating, find the flux of xi + yj + z k through the sphere of radius a and center at the origin. Take n pointing outward. 6B-2 Without calculation, find the flux of k through the infinite cylinder x2 + y2 = 1. (Take n pointing outward.) 6B-3 Without calculation, find the flux of i through that portion of the plane x+ y +z = 1 lying in the first octant (take n pointed away from the origin). 6B-4 Find JJsF . dS, where F = yj, and S = the half of the sphere x2 + y2 + z2 = a2 for which y 2 0, oriented so that n points away from the origin. 6B-5 Find JJsF .dS, where where F = z k , and S is the surface of Exercise 6B-3 above. 6B-6 Find JJsF . dS, where F = xi + yj + z k , and S is the part of the paraboloid z = x2 +y2 lying underneath the plane z = 1, with n pointing generally upwards. Explain geometrically why your answer is negative. xi +yj +zk 6B-7* Find JJsF dS, where F = x2 +y2 + z2 , and S is the surface of Exercise 6B-2. 6B-8 Find JJsF . dS, where F = yj and S is that portion of the cylinder x2 + y2 = a2 between the planes z = 0 and z = h, and to the right of the xz-plane; n points outwards. 6B-9* Find the center of gravity of a hemispherical shell of radius a. (Assume the density is 1, and place it so its base is on the xy-plane. 6B-lo* Let S be that portion of the plane -12x +4y +32 = 12 projecting vertically onto the plane region (x -+ y2 < 4. Evaluate a) the area of S2 E. 18.02 EXERCISES 6B-11* Let S be that portion of the cylinder x2 + y2 = a2 bounded below by the xy-plane and above by the cone z = J(x -a)2+ y2 . a) Find the area of S. Recall that dix = fisin(9/2). (Hint: remember that the upper limit of integration for the z-integral will be a function of 9 determined by the intersection of the two surfaces.) b) Find the moment of inertia of S about the z-axis. There should be nothing to calculate once you've done part (a). c) Evaluate /k z2 dS . 6B-12 Find the average height above the xy-plane of a point chosen at random on the surface of the hemisphere x2 + y2 + z2 = a2, z 2 0 . 6C. Divergence Theorem 6C-1 Calculate div F for each of the following fields a) x2yi +xyj +xzk b)* 3x2yzi+x3zj +x3yk c)* sin3xi +3ycos3xj +2xk 6C-2 Calculate div F if F = pn(x i + yj + z k), and tell for what value(s) of n we have div F = 0. (Use p, = xlp, etc.) 6C-3 Verify the divergence theorem when F = x i +yj +z k and S is the surface composed of the upper half of the sphere of radius a and center at the origin, together with the circular disc in the xy-plane centered at the origin and of radius a. 6C-4* Verify the divergence theorem if F is as in Exercise 3 and S is the surface of the unit cube having diagonally opposite vertices at (0,0,0) and (1,1,1), with three sides in the coordinate planes. (All the surface integrals are easy and do not require any formulas.) 6C-5 By using the divergence theorem, evaluate the surface integral giving the flux of F = xi + z2j + y2 k over the tetrahedron with vertices at the origin and the three points on the positive coordinate axes at distance 1from the origin. 6C-6 Evaluate JJsF .dS over the closed surface S formed below by a piece of the cone z2 = x2 + y2 and above by a circular disc in the plane z = 1; take F to be the field of Exercise 6B-5; use the divergence theorem. 6C-7 Verify the divergence theorem when S is the closed surface having for its sides a portion of the cylinder x2 + y2 = 1 and for its top and bottom circular portions of the planes z = 1and z = 0; take F to be 6C-8 Suppose div F = 0 and Sl and S2are the upper and lower hemispheres of the unit sphere centered at the origin. Direct both hemispheres so that the unit normal is "up", i.e., has positive k -component. a) Show that F dS, and interpret this physically in terms of flux. b) State a to an arbitrary closed surface S and a field F such that div F = 0.3 6. VECTOR INTEGRAL CALCULUS IN SPACE 6C-9* Let F be the vector field for which all vectors are aimed radially away from the origin, with magnitude llp2. a) What is the domain of F? b) Show that div F = 0. c) Evaluate where S is a sphere of radius a centered at the origin. /L F . dS, Does the fact that the answer is not zero contradict the divergence theorem? Explain. d) Prove using the divergence theorem that J& F . dS over a positively oriented closed surface S has the value zero if the surface does not contain the origin, and the value 47~if it does. (F is the vector field for the flow arising from a source of strength 47~ at the origin.) 6C-10 A flow field F is said to be incompressible if /L F . dS = 0 for all closed surfaces S. Assume that F is continuously differentiable. Show that F is the field of an incompressible flow u div F = 0 . 6C-11 Show that the flux of the position vector F = x i + yj + z k outward through a closed surface S is three times the volume contained in that surface. 6D. Line Integrals in Space 6D-1 Evaluate JcF . dr for the following fields F and curves C: a) F=yi+zj-xk; Cisthetwistedcubiccurve x=t, y=t2,z=t3 running from (0,0,0) to (1,1,1). b) F is the field of (a); C is the line running from (O,O, 0) to (1,1,1) c) F is the field of (a); C is the path made up of the succession of line segments running from (O,O, 0) to (1,0,O) to (Ill, 0) to (1,1,1). d) F=zxi+zyj+xk; Cisthehelix x=cost, y=sint, z=t, runningfrom (1,0,0) to (1,0,27r). 6D-2 Let F = xi + yj + …
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