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.
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface area - f(x, y) ds z =f(x, y) Lateral surface xy) As C: Curve in xy-plane Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface...
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.
Given the following vectors F=[y2, x2,x-z] and surface S: the triangle surface with vertices (0,0,1), (1,0,1), (1,1,1) in first octave. A. Evaluate the surface integral F(F) . dA B. Evaluate the surface integral VxF(F) dA C. Evaluate the line integral F() di where C is the curve enclosing the triangle. (Don't apply Green's theorem and integrate directly) Given the following vectors F=[y2, x2,x-z] and surface S: the triangle surface with vertices (0,0,1), (1,0,1), (1,1,1) in first octave. A. Evaluate the...
8. Find the area of the surface given by z - f(x, y) over the region R. f(x,y)- 42-x2-y2, R = {(x,y): x2 +y2 29 8. Find the area of the surface given by z - f(x, y) over the region R. f(x,y)- 42-x2-y2, R = {(x,y): x2 +y2 29
Exercise 4. Implicit differentiation (15 pts) Given z - xy + yz + y = 2 and z is a differentiable function in x and y. Then at (1,1,1) is: az дх a. 0 b. 1 1 C 2 d. d e. None of the above a. b. C. d. e. Exercise 6. Double integral in rectangular coordinates (10 pts+10 pts) Let I = S. secx dydx. 1) The region of integration ofl is represented by the blue region in:...
(4) Consider the surface z = x2+4y2+1. Suppose you are walking on this surface directly above a curve C in the xy-plane, where the parameterized curve is given by C (t)cost, y(t) sin t. Find the values of t for which you are walking uphil increasing z (Assume you are walking above the curve C in the direction of positive orientation The direction of positive orientation for the plane curve C is indicated by its tangent vectors.) (4) Consider the...
all questions are related and need help answering! rough the surface 4. o pm) What is the value of the flux of the vector field F(x,y)j+z ioriented with upward- pointing normal vector? (A) 0 (B) 2n/3 (C) π (D) 4T/3 (E) 2π Use Stokes, Theorem to evaluateⅡcurl F.dS, where F(x, y, z)-(x2 sin Theorem to evaluate Jceun F'.asS , where Fl.e)(', ») and 5. (5pts.) F,y, sin z, y', xy) and s is the part of the paraboloid : -...
The temperature at a point (x, y, z) is given by T(x, y, z) = 100e-x2 - 5y2 - 722 where Tis measured in °C and x, y, z in meters. (a) Find the rate of change of temperature at the point P12,-1, 3) in the direction towards the point (5, -3, 6). °C/m (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.
3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized by (u,v)-(ucos v, u sin v, hu) x2+y2 a at height h above the xy-plane Z = a V 0<vsa, OSvs 2n, and as is the curve parametrized by ē(f) =(acost,asint, h), 0sis27 as x2+ a 3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized...