Problem 1E: Explain how to compute the divergence of the vector field F = (f, g, h). Problem 2E: Interpret the divergence of a vector field. Problem 3E: What does it mean if the divergence of a vector field is zero throughout a region? Problem 4E: Explain how to compute the curl of the vector field F = (f, g, h). Problem 5E: Interpret the curl of a general rotation vector field. Problem 6E: What does it mean if the curl of a vector field is zero throughout a region? Problem 7E: What is the value of ( F)? Problem 8E: What is the value of u? Problem 9E: Divergence of vector fields Find the divergence of the following vector fields. 9.F = (2x, 4y, 3z) Problem 10E: Divergence of vector fields Find the divergence of the following vector fields. 10.F = (2y, 3x, z) Problem 11E: Divergence of vector fields Find the divergence of the following vector fields. 11.F = (12x, 6y, 6z) Problem 12E: Divergence of vector fields Find the divergence of the following vector fields. 12.F = (x2yz, xy2z,... Problem 13E: Divergence of vector fields Find the divergence of the following vector fields. 13.F = (x2 y2, y2 ... Problem 14E: Divergence of vector fields Find the divergence of the following vector fields. 14.F = (ex + y, ey +... Problem 15E: Divergence of vector fields Find the divergence of the following vector fields. 15.F=(x,y,z)1+x2+y2 Problem 16E: Divergence of vector fields Find the divergence of the following vector fields. 16.F = (yz sin x, xz... Problem 17E: Divergence of radial fields Calculate the divergence of the following radial fields. Express the... Problem 18E Problem 19E Problem 20E Problem 21E: Divergence and flux from graphs Consider the following vector fields, the circle C, and two points P... Problem 22E: Divergence and flux from graphs Consider the following vector fields, the circle C, and two points P... Problem 23E: Curl of a rotational field Consider the following vector fields, where r = (x, y, z). a.Compute the... Problem 24E: Curl of a rotational field Consider the following vector fields, where r = (x, y, z). a.Compute the... Problem 25E: Curl of a rotational field Consider the following vector fields, where r = (x, y, z). a.Compute the... Problem 26E: Curl of a rotational field Consider the following vector fields, where r = (x, y, z). a.Compute the... Problem 27E: Curl of a vector field Compute the curl of the following vector fields. 27.F = (x2 y2, xy, z) Problem 28E: Curl of a vector field Compute the curl of the following vector fields. 28.F = (0, z2 y2, yz) Problem 29E: Curl of a vector field Compute the curl of the following vector fields. 29.F = (x2 z2, 1, 2xz) Problem 30E: Curl of a vector field Compute the curl of the following vector fields. 30.F = r = (x, y, z) Problem 31E: Curl of a vector field Compute the curl of the following vector fields.... Problem 32E: Curl of a vector field Compute the curl of the following vector fields. 32.F=x,y,z(x2+y2+z2)1/2=r|r| Problem 33E: Curl of a vector field Compute the curl of the following vector fields. 33.F = (z2 sin y, xz2 cos y,... Problem 34E: Curl of a vector field Compute the curl of the following vector fields.... Problem 35E Problem 36E Problem 37E Problem 38E Problem 39E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 40E: Another derivative combination Let F = (f, g, h) and let u be a differentiable scalar-valued... Problem 41E: Does it make sense? Are the following expressions defined? If so, state whether the result is a... Problem 42E: Zero divergence of the rotation field Show that the general rotation field F = a r, where a is a... Problem 43E: General rotation fields a.Let a = (0, 1, 0), r = (x, y, z), and consider the rotation field F = a ... Problem 44E Problem 45E: Curl of the rotation field For the general rotation field F = a r, where a is a nonzero constant... Problem 46E: Inward to outward Find the exact points on the circle x2 + y2 = 2 at which the field F = (f, g) =... Problem 47E: Maximum divergence Within the cube {(x, y, z): |x| 1, |y| 1, |z| 1}, where does div F have the... Problem 48E Problem 49E: Zero component of the curl For what vectors n is (curl F)n = 0 when F = (y, 2z, x)? Problem 50E Problem 51E: Find a vector Field Find a vector field F with the given curl. In each case, is the vector field you... Problem 52E Problem 53E: Paddle wheel in a vector field Let F = z, 0, 0 and let n be a unit vector aligned with the axis of a... Problem 54E: Angular speed Consider the rotational velocity field v = 2y, 2z, 0. a.If a paddle wheel is placed in... Problem 55E: Angular speed Consider the rotational velocity field v = 0, 10z, 10y. If a paddle wheel is placed in... Problem 56E: Heat flux Suppose a solid object in 3 has a temperature distribution given by T(x, y, z). The heat... Problem 57E: Heat flux Suppose a solid object in 3 has a temperature distribution given by T(x, y, z). The heat... Problem 58E Problem 59E: Gravitational potential The potential function for the gravitational force field due to a mass M at... Problem 60E: Electric potential The potential function for the force field due to a charge q at the origin is... Problem 61E: Navier-Stokes equation The Navier-Stokes equation is the fundamental equation of fluid dynamics that... Problem 62E: Stream function and vorticity The rotation of a three-dimensional velocity field V = u, v, w is... Problem 63E: Amperes Law One of Maxwells equations for electromagnetic waves is B=CEt, where E is the electric... Problem 64E Problem 65E: Properties of div and curl Prove the following properties of the divergence and curl. Assume F and G... Problem 66E Problem 67E: Identities Prove the following identities. Assume that is a differentiable scalar-valued function... Problem 68E: Identities Prove the following identities. Assume that is a differentiable scalar-valued function... Problem 69E Problem 70E Problem 71E Problem 72E Problem 73E Problem 74E: Gradients and radial fields Prove that for a real number p. with r=x,y,z,(1|r|p)=pr|r|p+2. Problem 75E format_list_bulleted