Problem 1QC: Interpret the Divergence Theorem in the case that F=a,b,c is a constant vector field and D is a... Problem 2QC: In Example 3. does the vector field have negative components anywhere in the cube D? Is the... Problem 3QC: Draw unit cube D = {(x, y, z) : 0 x 1, 0 y 1, 0 z 1} and sketch the vector field F=x,y,2z on... Problem 1E: Review Questions 1.Explain the meaning of the surface integral in the Divergence Theorem. Problem 2E: Interpret the volume integral in the Divergence Theorem. Problem 3E: Explain the meaning of the Divergence Theorem. Problem 4E: What is the net outward flux of the rotation field F = 2z + y, x, 2x across the surface that... Problem 5E: What is the net outward flux of the radial field F = x, y, z across the sphere of radius 2 centered... Problem 6E: What is the divergence of an inverse square vector field? Problem 7E: Suppose div F = 0 in a region enclosed by two concentric spheres. What is the relationship between... Problem 8E: If div F 0 in a region enclosed by a small cube, is the net flux of the field into or out of the... Problem 9E: Verifying the Divergence Theorem Evaluate both integrals of the Divergence Theorem for the following... Problem 10E: F = x, y, z; D = {(x, y, z): |x| 1, |y| 1, |z| 1} Problem 11E: Basic Skills 912.Verifying the Divergence Theorem Evaluate both integrals of the Divergence Theorem... Problem 12E: F = x2, y2, z2; D = {(x, y, z): |x| 1, |y| 2, |z| 3} Problem 13E: Rotation fields 13.Find the net outward flux of the field F = 2z y, x, 2x across the sphere of... Problem 14E: Rotation fields 14.Find the net outward flux of the field F = z y, x z, y x across the boundary... Problem 15E: Find the net outward flux of the field F = bz cy, cx az, ay bx across any smooth closed surface in... Problem 16E: Rotation fields 16.Find the net outward flux of F = a r across any smooth closed surface in 3 where... Problem 17E: Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields... Problem 18E: Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields... Problem 19E: F = x, 2y, z; S is the boundary of the tetrahedron in the first octant formed by the plane x + y + z... Problem 20E: Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields... Problem 21E: F = y 2x, x3 y, y2 z; S is the sphere {(x, y, z): x2 + y2 + z2 = 4}. Problem 22E: Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields... Problem 23E: Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields... Problem 24E: Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields... Problem 25E: Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward... Problem 26E: Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward... Problem 27E: Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward... Problem 28E: Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward... Problem 29E: F = x2, y2, z2); D is the region in the first octant between the planes z = 4 x y and z = 2 x y. Problem 30E: Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward... Problem 31E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 32E: Flux across a sphere Consider the radial field F = x, y, z and let S be the sphere of radius a... Problem 33E: Flux integrals Compute the outward flux of the following vector fields across the given surfaces S... Problem 34E: Flux integrals Compute the outward flux of the following vector fields across the given surfaces S.... Problem 35E: Flux integrals Compute the outward flux of the following vector fields across the given surfaces S... Problem 36E: Radial fields Consider the radial vector field F=r|r|p=x,y,z(x2+y2+z2)p2. Let S be the sphere of... Problem 37E: Singular radial field Consider the radial field F=r|r|=x,y,z(x2+y2+z2)1/2. a.Evaluate a surface... Problem 38E: Logarithmic potential Consider the potential function (x,y,z)=12ln(x2+y2+z2)=ln|r|, where r = x, y,... Problem 39E: Gauss Law for electric fields The electric field due to a point charge Q is E=Q40r|r|3, where r = x,... Problem 40E: Gauss Law for gravitation The gravitational force due to a point mass M at the origin is... Problem 41E: Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F... Problem 42E: Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F... Problem 43E: Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F... Problem 44E: Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F... Problem 45E: Heat transfer Fouriers Law of heat transfer (or heat conduction) states that the heat flow vector F... Problem 46E: Inverse square fields are special Let F be a radial field F = r/|r|p, where p is a real number and r... Problem 47E: A beautiful flux integral Consider the potential function (x, y, z) = G(p), where G is any twice... Problem 48E: Integration by parts (Gauss' Formula) Recall the Product Rule of Theorem 14.11: (uF) = uF + u(F).... Problem 49E Problem 50E Problem 51E: Greens Second Identity Prose Greens Second Identity for scalar-valued functions u and v defined on a... 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