Prove: If
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- 4. Show that the following vector fields F are conservatives and find ALL f such that F = Vf. = (a) (b) (2xy+z³)i+x²j+3xz²k i+ y)² X F(x, y, z) = F(x, y) = {(x, y) = R² | x > y}. (x Y X : - y)2j for all (x, y, z) € R³. for all (x, y) E U :=arrow_forwardLet (f (u, v,w) and (ü = u(x, y, z), v = v(x, y, z), w = w(x, y, z) be differentiable functions) prove that grad(f) = Vf = fuVu + fVv + fwwarrow_forward(a) Let f(x, y) = P(x, y)i +Q(x, y)j be a vector field, where P and Q are continuous differentiable functions. Let C be a smooth curve parameterized by x= x(t), y = y(t) where ast≤b with position vector r(t) = x(t)i + y(t)j. Show that ff.dr=-ff.dr C Hence prove that ff.dr = 0 (b) Use divergence theorem to calculate the surface integral fF.dS; that is, calculate the flux S of F across S. F(x, y, z) = (y² cos z + xy²)i +(x³e¯²)j +[sin(3y) + 1x²z]k, S is the surface of the solid bounded by the paraboloid z = x² + y² and the plane z = 6arrow_forward
- 2. For the vector field F (x, y, z) = (x²y, xyz, −x²y²) F(x, y, z)=(x2y,xyz,-x2 y 2 ), find: (a) div (F)div (F) (b) curl (F)curl(F) (c) div (curl (F))arrow_forward3.5arrow_forward2. Let U be a vector function of position in R³ with continuous second partial deriva- tives. We write (U₁, U2, U3) for the components of U. (a) Show that V (U • U) – Ü × (▼ × Ü) = (U · ▼) Ū, (V where ((UV) U), = U₁ (b) We define the vector function of position, by setting = V × . If the condition V U = 0 holds true, show that ▼ × ((Ū · V) Ű) = (Ũ · V) Ñ - (Ñ· ▼) ū. (Hint: The representation of (UV) U from the first part might be useful.)arrow_forward
- Explain brieflyarrow_forwarda) Show that F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j is the gradient of some function f. Find f b) Evaluate the line integral ʃC F dr where the vector field is given by F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j and C is the curve on the circle x 2 + y 2 = 9 from (3, 0) to (0, 3) in a counterclockwise direction.arrow_forwardREFER TO IMAGEarrow_forward
- 7. Show that if the vector field F = Pi + Qj + Rk is conservative and P,Q, R have continuous first-order partial derivatives, then %3D de ӘР ƏR aQ ƏR dy az dz ду Hint: Recall that Clairaut's theorem states that if h(x, y, z) is a function whose second partials are all continuous, then hry = hyæ, haz hzz, and hyz = hzy.arrow_forwardThe nabla formulas corresponding to the derivative rule of income are gVf+fVg, (Vƒ). F + f(V.F), (Vf) x F + f(V x F). ▼(fg) V. (fF) ▼ x (fF) = = = Justify the middle one (e.g. in the case of n = 3). Hint: The coordinate functions of the vector field fF are fF₁ etcarrow_forward
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