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EBK CALCULUS EARLY TRANSCENDENTALS SING
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- Consider the vectorial V=2+ŷ+ 2 z . . function=z²+x²y + y²2 and the gradient operator Please explicitly evaluate Vxarrow_forwardLet f = f(x, y, z) be a sufficiently smooth scalar function and F = Vƒ be the gradient acting on f. Which of the following expressions are meaningful? Of those that are, which are necessarily zero? Show your detailed justifications. (a) V· (Vf) (b) V(V × f) (c) V × (V · F) (d) V. (V × F)arrow_forwardShow that X'= B(X, K) is a Banach space.arrow_forward
- (a) Let u be a function of R2 such that ди ди an(x, y) = (x, y); for all(x, y) ∈ R2 ду then prove that au (x,y") ду u(x, y) – u(y, x) = (x-y) (x*, y*) + (y-x) ди Әх for some point (x*, y*) ∈ R2 (b) Evaluate the general solution of partial differential equation (x−y)ux+(y−x−u)uy =uarrow_forward2. 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_forwardB) Find the work done by the conservative vector field F (r, y, z) = yzi+xzj+ryk along a smooth curve C from the point (-1,3, 8) to (1,6, –4).arrow_forward
- 2. 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_forwardExplain brieflyarrow_forwardLet F₁ (x, y, z) = (y, x − z² +2,−2yz) and F₂ (x, y, z) = (x − z², −y + 2, 2yz). (a) Find the curl of each vector field. (b) Which of these two vector fields is conservative? Find a potential function for the one that is. (c) Let F be either F₁ or F2 (whichever you want), and let C denote the curve shown in the figure which starts at the point (0, 1,0), stays above the circle x² + y² = 1 and spirals upwards in a clockwise direction to the point (0,1,1). Jo Find the exact value of the line integral: F. dr.arrow_forward
- 6 step by steparrow_forward3. Let V denote the vector space of all functions ƒ : R" → R, equipped with addition + : V × V → V defined f: via (f+g)(x) = f(x) + g(x), x = R", and scalar multiplication : Rx V → V defined via (A. f)(x) = \f(x), ● x ER". Now let W = {f: R" → R: f(x) = ax + b for some a, b ≤ R}, i.e. the space of all linear functions R" → R. (a) Show that W is a subspace of V. (You may assume that V is a vector space). (b) Find a basis for W. You should prove that it is indeed a basis.arrow_forwardShow that V x (V.f) = 0 for any vector field f.arrow_forward
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