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(x2 + y2 + z?)1/2, and e, = r1(x, y, z) is the unit radial vector....
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2...
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2 + z2)1/2" a. Evaluate a surface integral to show that SsFonds = 4ta?, where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a)...
(,y,) dS, where f(,y,) = z'yz2 and S is the part ys 4 1. Parametrize, but do not evaluate, +y of the graph of...
(,y,) dS, where f(,y,) = z'yz2 and S is the part ys 4 1. Parametrize, but do not evaluate, +y of the graph of z over the rectangle -2 S rs3 and 0 2. Parametrize, but do not evaluate, F.n dS, where F (y,-r,z) and S is the sphere of radius 2 centered at the origin. Math 224 3. Calculate le ayz dS where S is the part of the cone parametrized by 0sus1,0svs r(u, v)(ucos v, usin v, u),...
A B C Parametrize, but do not evaluate, //f(x, y, z) ds, where f(x, y, z) 2y22 and S is the part , where J(,y,) 3...
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B
C
Parametrize, but do not evaluate, //f(x, y, z) ds, where f(x, y, z) 2y22 and S is the part , where J(,y,) 3 3 and 0 Sys4 of the graph of z2 over the rectangle -2 s . Parametrize, but do not evaluate, F.n ds, where F (,-,z) and S is the sphere of radius 2 centered at the origin. Calculate JJs xyz dS where S is the part of the cone parametrized by r(u, u) (ucos...
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the...
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the surface S be the part of the sphere x2 + y2 + x2 = 4 that lies above the plane 2=1 and be oriented downwards. (a) Find the divergence of F. (b) Compute the flux integral SS. F . ñ ds.
5. Calculate the surface area of the portion of the sphere x2+y2+12-4 between the planes z-1 and z ะไ 6. Evaluate (xyz) dS, where S is the portion of the plane 2x+2y+z-2 that lies in the first oc...
5. Calculate the surface area of the portion of the sphere x2+y2+12-4 between the planes z-1 and z ะไ 6. Evaluate (xyz) dS, where S is the portion of the plane 2x+2y+z-2 that lies in the first octant. 7. Evaluate F. ds. a) F = yli + xzj-k through the cone z = VF+ア0s z 4 with normal pointing away from the z-axis. b) F-yi+xj+ek where S is the portion of the cylinder+y9, 0szs3, 0s r and O s y...
where g is a function of one variable 16. Suppose that f(x,y,z)= g(V x2 + y2...
where g is a function of one variable 16. Suppose that f(x,y,z)= g(V x2 + y2 + such that g(3) = 4. Evaluate ſyf(x,y,z)ds' where S is the sphere x² + y2 +z2 =9.
across Let F = (x i+yj +z k'). What is the outward flux of F (x2...
across Let F = (x i+yj +z k'). What is the outward flux of F (x2 + y2 +22)8 a sphere of radius R> 0 centered at the origin?
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisph...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
14. (1 pt) Calculate the flux of the vector field = 6r. where ř = (x,...
14. (1 pt) Calculate the flux of the vector field = 6r. where ř = (x, y, z), through a sphere of radius 3 centered at the origin, oriented outward. Flux =
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2 + z2)1/2" a. Evaluate a surface integral to show that SsFonds = 4ta?, where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a)...
(,y,) dS, where f(,y,) = z'yz2 and S is the part ys 4 1. Parametrize, but do not evaluate, +y of the graph of z over the rectangle -2 S rs3 and 0 2. Parametrize, but do not evaluate, F.n dS, where F (y,-r,z) and S is the sphere of radius 2 centered at the origin. Math 224 3. Calculate le ayz dS where S is the part of the cone parametrized by 0sus1,0svs r(u, v)(ucos v, usin v, u),...
A
B
C
Parametrize, but do not evaluate, //f(x, y, z) ds, where f(x, y, z) 2y22 and S is the part , where J(,y,) 3 3 and 0 Sys4 of the graph of z2 over the rectangle -2 s . Parametrize, but do not evaluate, F.n ds, where F (,-,z) and S is the sphere of radius 2 centered at the origin. Calculate JJs xyz dS where S is the part of the cone parametrized by r(u, u) (ucos...
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the surface S be the part of the sphere x2 + y2 + x2 = 4 that lies above the plane 2=1 and be oriented downwards. (a) Find the divergence of F. (b) Compute the flux integral SS. F . ñ ds.
5. Calculate the surface area of the portion of the sphere x2+y2+12-4 between the planes z-1 and z ะไ 6. Evaluate (xyz) dS, where S is the portion of the plane 2x+2y+z-2 that lies in the first octant. 7. Evaluate F. ds. a) F = yli + xzj-k through the cone z = VF+ア0s z 4 with normal pointing away from the z-axis. b) F-yi+xj+ek where S is the portion of the cylinder+y9, 0szs3, 0s r and O s y...
where g is a function of one variable 16. Suppose that f(x,y,z)= g(V x2 + y2 + such that g(3) = 4. Evaluate ſyf(x,y,z)ds' where S is the sphere x² + y2 +z2 =9.
across Let F = (x i+yj +z k'). What is the outward flux of F (x2 + y2 +22)8 a sphere of radius R> 0 centered at the origin?
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
14. (1 pt) Calculate the flux of the vector field = 6r. where ř = (x, y, z), through a sphere of radius 3 centered at the origin, oriented outward. Flux =
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