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4. (6 pts) Compute the following quantities: (a) Vf where f = 2xy – yz +...
(6 pts) Compute the following quantities: (a) Vf wheref 2xy -yz+ x2 cos(z) (b) V Fi where F1 = 2xî + yzy - z2i (c) Vx F2 where F2 = e~jzg
Calculate v? f where f = 2e" +5x^y+cos(yz) IV-f=V (vf), the divergence of the gradient of fl [2e' +10y-z cos(yz) – y cos(yz)]
you can skip #2
Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u where f(r,y,) = =- +22 2. Consider the vector field F(E,) = (a,y) Compute the flow lines for this vector field. 3. Compute the divergence and curl of the following vector field: F(x,y,)(+ yz, ryz, ry + 2)
Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u...
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...
number 4 parts a b and c
The symbols Vf, V.. bols vf, V az oy F, and V x F are defined by Of = grad f 7. F = div F VXF = curl F After eq. (3.12) is memorized, formulas (3.13) through (3.15) venient ways of remembering the expressions for gradient, diverge just operate with V as though it were a vector. Henceforth, we breviations frequently. EXERCISES 1. If f(x,y,z) = x²y + z, what is f(2,3,4)?...
F(x,y,z)= (y² +e",2xy +z sin y, -cos y) is a gradient vector field. Compute Sc F. dr where C=C UC2, C, is the curve y=x*, z =0 from (0,0,0) to (1,1,0) and C, is the straight line from (1,1,0) to (2,2,3).
F(x, y,z)=(y2 +e", 2xy + z sin y, cos y) is a gradient vector field. Compute Sc F. dr where C=GUC,, C işthe curve y = x^, z = 0 from (0,0,0) to (1,1,0) and C, is the straight line from (1,1,0) to (2,2,3)
4 (16 pts). F = (4 + 2xy)i + (x2 – 2y2)). (1) Show that Ě is conservative and then find a function f such that of = Ē. (2) Use the function f to evaluate Sc F • dř, where the curve C:r(t) = et cos(t)i + etsin(t)1,0 St Sil.
96. Consider a vector field F(x, y, z) =< x + x cos(yz), 2y - eyz, z- xy > and scalar function f(x, y, z) = xy3e2z. Find the following, or explain why it is impossible: a) gradF (also denoted VF) b) divF (also denoted .F) c) curl(f) (also denoted xf) d) curl(gradf) (also denoted V x (0f) e) div(curlF) (also denoted 7. (V x F))
Question 14 7 pts Consider the line integral F. dr where REC IND РІ. F(x, y, z) = i + (x+yz)j + (xy – z)k and C is the boundary of the plane 2 + y + z = 4 in the first octant, oriented in the counterclockwise direction when viewed from above. the following double integrals is equivalent to this line Using Stokes' Theorem, which integral? °6964 (3 - 2z+1) du dz (2x + y) dy da Question 12...