4. Evaluate both sides of Green's theorem, for: F = (-x²y,xy²) where C is shown in the following figure. y a C R a

Q: Show that if F = xi + yj + zk, then ∇ * F = 0.

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Q: Find f, and f, and evaluate each at the given point. y' f(x, y) = e*y2, (In 4, 2) %3D

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Q: 1. Express the followings in the form of x + iy where r, y E R.(choose one.) (a) 금+ 부 (b) ( + ) ( –…

A: Rationalise  all the term and calculate to get the final answer.   Actually as per the barteleby…

Q: Find f, and f, and evaluate each at the given point. 4xy f(x, у) %3D (1, 1) V 2x2 + 7y2' f,(x, y) =…

A: This question is related to partial derivatives as solved in step 2.

Q: if f(z) = (ax – x³ + 3xy³)+ (by – 3x²y+ y') is anal ytic , then the value of a and is

A: We have , f(z) = ax - x3 + 3xy2 + iby - 3x2y + y3 We have given that ,  f  is analytic function.…

Q: given f (z) = 32² – 6z + j3, what is the value of u(3, 2)

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Q: Q : If ƒ,g:R→R are defined respectively by f (x) = x² + 3x + 1, g(x) = 2x – 3 find (i) fog %3D

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Q: So, which of the following is the solution functiony(t) ? a) y(t)= sinh2t = e* %3D b) y(t) = sin2t –…

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Q: 2.Evaluate . xy+ds where C is the right half of x2 + y? = 16

A: Integrate with use parametric form

Q: where C is rE by the lines X

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Q: Given f(x, y, z) = 2e(z+y), Find fy(0,},-4).

A: here we have to find partial derivative of a function w.r.t. 'y' and here we also use chain rule.

Q: Let T: P2(IR) → P2 (R) be defined by T(p(x)) = p(x) – 3p'(x). Given that %3D - T(1+x + x?) = a + bx…

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Q: Q/ z = 1+ f(a) and a = ry. Show that: r-y =

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Q: now if we take F (x, y) = (ax, ay), where a is any positive constan %3D

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Q: Given that f"(x) = cos(x), f'(T/2) = 8 and f(T/2) = 7, find: %3D f'(z) =O %3D f(2) =U

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Q: If Z = x.n. f .f (**²), then show that: X. Zx + y.Zy = y

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Q: _1. Differentiate the following w.r.to x (i) V(x? + 1)3 .3,-2x (ii) + In (3x + 1)

A: Using derivative ,

Q: Find f, and f, for f(x, y) = 10(9x – 8y+ 12)°. fx(Xx.y) = f,(x.y) D

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Q: 17 and g(x) = 'Vx, find (f o g)(x) and (g o f)(x). Then determine wh %3D fo g)(x)?

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Q: 7. If DE %3D 11x + 15 and FG 32x - 27, find x. %3D F D H.

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Q: Q3 Given that w = x + 2y + z²,x = = ,y = r2 + In s, z = 2r X%3D %3D aw aw and ar Find as

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Q: Find f, and f and evaluate each at the given point. 2xy x, - V+ ay (1, 1) K, ) - Sx, ) - (1, 1) - 29…

A: The solution of the problem is given in the next steps.

Q: (1. show that a) { f} = jw Flw) c)

A:

Q: Consider f(x, y) = x³ + 2ry + y. Compute the following: (b)

A: Answer and explanation is given below.. Simply do partial derivative with respect to x and y.

Q: If f(x, y) = 8xe2y + 8y Inx, then is given by:

A: The given function is: f(x,y)=8xe2y+8y ln x Now, ∂f∂x=∂∂x8xe2y+8y ln x       =8e2y+8yx

Q: given f (z) = (z + 1)° – 32' what is the value of u (금, )

A: Topic:- complex number

Q: Evaluate F. dr. F(x, y) = xi + yj C: r(t) = (3t+1)i + tj, 0sts 1

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Q: Given f(x,y,z) = x³yz². Find fy(1,1,1)

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Q: 2. Let f(x,y)=xe" and find a) fx, fyr fy

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Q: If f(x, y) = (2x? + 8y²)í, then f_(0, 0) = %3D A) 0 В) 1 C) 4 D) 2 E) Does not exist

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Q: x2 B) If w = x".f then show that: %3D x. Wx + y. Wy = nw.

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Q: Let f (2) = x² + iywhere x and y are the real and imaginary parts, respectively of z and C is a…

A: The given problem is to find the value of Re[f(z2)], where given z2 = 8+7i and function f(z)=x2+iy2…

Q: Given that f"(x) = cos(x), f'(a/2) = 13 and f(t/2) = 14, find: %3D %3D f'(x) = f(x) =

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Q: . Evaluate F dr, where F(x, y) = ryi+z'y²j, and C is given by r(t) = ti+tj. 0sts1.

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Q: Q3: 1f f(x, y) = 4xCy5 + 6x*y* .Prove that: f(x,y) * fxxyy fxx * fyy = 1

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Q: Q3. If F = x'a, + y'a, + (z² – 1) a̟, find F· dS, where S is defined by r= 2, 0 < z < 2,0 < ¢ < 2r.

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Q: ニメ Find the eg uaton (x) 4a| %3D 3tan x when fo f the tor

A:

Q: If r(t) = (V10 – t, In(t – 1) - then the domain of r is

A: Domain = (1, 10]

Q: 2. Show that the function u = p(ry) + /TY p(y/x) provides the equation ‚Pu dy² y² = 0. %3D

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Q: y = f(x) ナ

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Q: If f(x) = Rsin TCX +S,f' 12 and 2R ,then the constants R %3D 2 0. and S are respectively. 1

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Q: If F (x, y, z) = e^xz i + 3xe^y j − e^yz k, then find ∇.(∇ × F)

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Q: Q3) If f () = 3y5+2x5+4x2y³ prove the following %3D x²y3 af(x,y) af (x,y) 6(x + y). %3D əx ду

A: Given: fyx,xy=3y5+2x5+4x2y3x2y3

Q: Let M,(t) = 1/(1 - 3t) for x > 0. a. Find E(X). b. Find V(X).

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Q: Given that f(x) = c/x with x = 1,2,3. Find M(t)

A: It is given that, f(x) = c/x with x = 1,2,3. To find M(t), first of all, we have to find c which can…

Q: given f (z) = (z +1)° - 3z' what is the value of u

A:

Q: For u(x, t) = f (x + ct) + g (x – ct), where c is a constant, find ux, ut, uxt . Show your work…

A: Given, u(x, t) = f (x + ct) + g (x – ct)

Q: 3. Suppose that ƒ € C'(B1,Rk) satisfies ||Df(0)|| = }. Prove that 3 6 > 0 such that %3D |f(1x) –…

A: Given: Assume that f∈C1B1,ℝk satisfies D f(0)=12, Now e need to prove that ∃δ&gt;0 as a result…