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Problem 5. Let F(r,y) (e-v-v sinzy) ?-(ze-s + z sin zyj (1) Show that F is a gradient field. (2) Find a potential function f for it (3) Use the potential function f to evaluate F-ds, where x is the path x(t) = (t,t2) for 0sts1. (NO credit for any other method.)
Let f(x,y,z) = xy + z-5,x=r +2s, y = 2r - sec(s), z = s Then I is: ar a. r - sec(s) b. sec(s) c. r+s+sec(s) d. 4r + 4s - sec(s) a. b. C. Given zº – xy + y2 + y2 = 2 and z is a differentiable function in x and y. Then at (1,1,1) is: дх a. 0 b. 1 c. d. e. None of the above o a. o b. ♡ C. o d.
aw 4. Find when (r, s) = (1, -1) if w = (2+y+z)?, r=r-s, y = cos(r +s), z = sin(r +s). ar 5. Find the directional derivative of f(x, y, z) = 3x² + yz + 2yz? at P(1,1,1) in a direction normal to the surface x2 – y + z2 = 1.
2. You are given the following multivariate PDF (x, y, z) E S X,Y,2(x, y,z)=)4m 0 else where S-((x, y, z) 1x2 + y2 +#51). (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the...
(P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C (P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C
Let F(x, y, z) = x2y3 + y 2 sin(π z) /π + z2ex-1 a) Find the equation of the tangent plane to the graph of the function z = z(x, y) at the point (x, y) = (1, 1), if z satisfies the equation F(x, y, z) = 2 with z(1, 1) = 1. b) At the point P(1, 1, 1), determine in which of the two directions ~u = h−4, 3, 0i or ~v = h−3, 0, 4i...
This is a MATLAB question so please answer them with MATLAB steps. Let f(z) = V3z sin(#) and P(z) =r-x-1. 1. Find f(e) 2. Find the real solution(s) to Px) 0. Hint: use the roots command. 3. Find the global minimum for f(x). Hint: plot f over [0,2] 4. Solve f()P. Hint: plot f and P over [0,21. 5. Find lim,→0+ f(x). Hint: make a vector hi make a table [x + h; f(x + h)]". 6. Find '(In 2)....
11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
= r.Cos (0), y r sin(0), and zr0 Let x.y,z)=x y+y zxz, where x 3-where w(r,0) = u(x(r,0),y(r,0),2(r,0)) Owr.0) for r= 1, 0 = д0 (r,0) and дr Evaluate 2 = r.Cos (0), y r sin(0), and zr0 Let x.y,z)=x y+y zxz, where x 3-where w(r,0) = u(x(r,0),y(r,0),2(r,0)) Owr.0) for r= 1, 0 = д0 (r,0) and дr Evaluate 2
-n ', S Let f(x,yZFz2_xy. Let v=<1,1,1>. Let point P=<2,1,3> a. Compute gradient of fx,y,z) b. If the contours are far apart, is the length of the gradient large or small? Answer: Explain! What MATLAB command is used to draw the gradient vectors? Answer: - c. Compute the directional derivative in the direction of v. d. Compute the equation of the tangent plane to f(x,y,z) at the point P. e. Use the chain rule to compute r if x t2,...