Let S be the parabolic arch given by r(u,v) = (u+v, u-v, 4-u²) where (u, v) € [−2, 2] × [0, 1]. Compute the normal vector xr Here is a picture of this surface. Draw (in several representative places) the direction of the normal vectors that you computed in the previous part of the problem. Set up and evaluate the flux integral of F = j across S. I

Let S be the parabolic arch given by r(u, v) = (u+v, u−v, 4-u²) where (u, v) € [−2, 2]×[0, 1]. Compute the normal vector × ди Iv' Ər Here is a picture of this surface. Draw (in several representative places) the direction of the normal vectors that you computed in the previous part of the problem. Set up and evaluate the flux integral of F = xj across S.

−Y+z5 Let F (x, y, z) := x+yz7 3+2⁹ let ʼn be the outward unit normal vector. Evaluate ſf (curl F) · ndS. (Use syntax like 2*pi/3 .) Let S be the upper half of the sphere of radius 2 centered at the origin. At each point of S

Find a vector function that represents the curve of intersection of the two surfaces. The cone y = V = v 2x2 +9z2 and the plane y = 5+3z r(t) = ( 1 t2 2' 15 6. 1 r(t) = (t, 1 t2 2' 15 6. 1 5 r(t) = (! + 구 t, 15 %3D 6. r(t) = (t, - + 2' 15 6. r(t)%3D = ( t, ť + 15 6.

Let S be the parabolic arch given by r(u, v) = (u+v, u−v, 4−u^2) where (u, v) ∈ [−2, 2]×[0, 1]. Compute the normal vector ∂r/∂u × ∂r/∂v . Here is a picture of this surface. Draw (in several representative places) the directionof the normal vectors that you computed in the previous part of the problem. Set up and evaluate the flux integral of F = x j across S.

Find a vector normal to the surface x2 + yz = 5 at (2, 1, 1).

Find an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 – cos (v)) sin (u), z = sin (v) at the image of a point (u, v) for –x < v < a and -a < u < a. (Write your solution using the form (*,*,*). Use symbolic notation and fractions where needed.) no = Identify this surface. ellipsoid hyperboloid torus cylinder

What is the principal unit normal vector for the curve: r(t) = (3t + 1)i + e'j .at the point P = (1,1)? 3i – j V10 Ob. 3i +j V10 Oc. V10 3/10 i+ 10 10 od There is no unit normal vector for this curve at this point. V10 3V10 -j 10 10

2. Calculate the gradient vector Vf of the function f (x, y) = x² – x + y - x²y - 2y2 at the point (2,1) and sketch it on the attached contour plot (you can save the picture, open in photo editor and use drawing tools). Explain in one paragraph (about 200-300 words) the meaning of the gradient vector Vf(2,1), negative gradient vector -Vf(2,1).

Let the temperature in degrees at the point (X,y,z), with distances measured in cm, be T(X,y,z)=5x – 4y +3z. Let 9 be the real number such that the rate at which the change in temperature at (1,0,3) per unit change in the distance travelled in the direction of the vector (1,q,1> is 4 /cm. Find . (Note that the "direction" of a vector is always a unit vector "pointing the same way.") none of the other answers -7/40 17/56 1/12 1/2