6 Vector Integral Calculus in Space 6A Vector Fields in Space 6A 1 Describe geometrically the following vector fields a xi yj zk P b x i z k 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 x i y j and center at the origin Take n pointing outward zk through the sphere of radius a 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 y j and S the half of the sphere x2 for which y 2 0 oriented so that n points away from the origin y2 z2 a2 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 x i y j 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 6B 7 Find JJsF dS where F xi yj zk and S is the surface of Exercise 6B 2 x2 y2 z2 6B 8 Find JJsF dS where F y j and S is that portion of the cylinder x2 y 2 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 the plane region x y2 4 Evaluate a the area of S 4y 32 12 projecting vertically onto E 18 02 EXERCISES 2 6B 11 Let S be that portion of the cylinder x2 xy plane and above by the cone z J x a 2 y2 y2 a2 bounded below by the a Find the area of S Recall that d i x 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 d S 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 x 2 y i x y j x z k b 3 x 2 y z i x 3 z j x 3 y k 6C 2 Calculate div F if F pn x i div F 0 Use p xlp etc c sin3xi 3 y c o s 3 x j 2 x k yj z k and tell for what value s of n we have 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 a t 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 x i z2j y2 k over the tetrahedron with vertices a t the origin and the three points on the positive coordinate axes at distance 1 from 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 1 and 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 b State a F dS and interpret this physically in terms of flux to an arbitrary closed surface S and a field F such that div F 0 6 VECTOR INTEGRAL CALCULUS IN SPACE 3 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 L F dS where S is a sphere of radius a centered a t the origin Does the fact that the answer is not zero contradict the divergence theorem Explain J d Prove using the divergence theorem that 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 4 7 a t 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 y j 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 Jc F dr for the following fields F and curves C a F y i z j x k from 0 0 0 to 1 1 1 Cisthetwistedcubiccurve x t y t 2 z t 3 running b F is the field of a C is the line running from O …
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