vect_intgrl_calc

# vect_intgrl_calc - MIT OpenCourseWare http/ocw.mit.edu...

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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 + y j + 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 JJs F . dS, where F = y j , and S = the half of the sphere x2 + + z2 = a2 for which y 2 0, oriented so that n points away from the origin. 6B-5 F . dS, where where F = z k , S is the surface of Exercise 6B-3 above. 6B-6 F . dS, where F = xi + y j + z k , 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. 6B-7* F dS, where F = + + , S is the surface of Exercise 6B-2. y2 6B-8 F . dS, where F = y j and S is that portion of the cylinder x2 + = 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 + + 32 = 12 projecting vertically onto the plane region (x - + < 4. Evaluate a) the area of S
2 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 + . 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 + + 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 + y j + 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 + y j + 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.

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vect_intgrl_calc - MIT OpenCourseWare http/ocw.mit.edu...

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