8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z cos y, xz sin y, x cos y). a) Determine which vector field above is conservative. Justify. Foly = fjol so, <ea sin(J2), 20% cos(82), y acos (92)) Conservative. b) For the vector field that is conservative, find a function f such that F - Vf. Lxelsing2, zetos yea, yet cosy 2 c) Use the Fundamental Theorem of Line Integrals to find the work...
Consider F and C below. F(x, y, z) = yz i + xz j + (xy + 10z) k C is the line segment from (3, 0, -3) to (4, 4, 1) (a) Find a function f such that F = Vf. f(x, y, z) = (b) Use part (a) to evaluate [s vf. dr along the given curve C.
. Let F(x, y, z) = ze+7 +(+ Iny)ī – (z2 + arctan y)k. (a) Calculate the curl of F, or 7 x 7. (b) Calculate the divergence of F, or VF.
Let f(x, y, z)=x2-7xy +32 Find Vf. Vr= (Type your answer in terms of i, j, and k.) This Question: 2 pts -1/2 Find the gradient of f(x,y,z) = (2+2+2) +In (xyz) at the point (1.-2.-2). -OOO (Type simplified fractions Enter your reach of these Parmak için buraya yazın
b) Consider the surface in R3 described by f(x,y,z) = 2x²y3 + z + ye*2 = 9 (i) Find Vf(x,y,z). [3 marks] (ii) Verify that (2,1,0) is a point on this surface. Find the cartesian equation of the tangent plane to this surface at the point (2.1,0). [5 marks]
Find Vf at the given point. f(x,y,z) = x2 + y3 – 322 + z Inx, (1,1,4) Vf|(1,1,4) = i+ )j + (O)k (Simplify your answers.)
Find Vf at the given point. f(x,y,z) = x3 + y3 – 322 + z Inx, (1,5,5) Vf|(1,5,5) = Di+(\)j + ()k (Simplify your answers.)
7. The vector field F =< 3x2z In y + ze+2 +20, - 3y?, x° In y + ce2 +423 > is conservative. Find a potential function f(x, y, z) such that F=Vf. Y
Consider F and C below. F(x, y, z) = yze?i + e'?j + xyek, C: r(t) - (t? + 1)i + (t? - 1)j + (t– 3t)k, Osts3 (a) pind a function f such that F – Vf. f(x, y, z) (b) Use part (a) to evaluate F. dr along the given curve C.
Find the directional derivative of f at p in the direction of a. f(x,y,z)=xy+z^2; P(2,-2,2);A=i+j+k