Gradients and radial fields Prove that for a real number p, with r= (4, y. 2), vE -pr p+2

Q: Exercise.1 Find the roots for V8 and V-1

A: According to our guidelines, we can answer only the first question, the rest can be reposted.

Q: Find div F and curl F if F(x,y, z) = x°i – 11j + yzk. - div F= curl F=

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Q: f(x) = x² + sin z %3D is an antiderivaive of Athat satisfies F(x) = 0.| %3D

A: Given;- fx=x2+sin x   Solution:- ⇒∫x2+sin x dx ⇒∫x2 dx+∫sin x dx ⇒x33-cos x ⇒x33-cos x+C…

Q: A company has $300,000 to spend on two products. Suppose that the Utility derived by the company…

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Q: An image is projected onto a wall, and the irradiance of the light on the wall is (3) given by f(x,…

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Q: use linear approximation to give an estimate for the value of g (0.97, 2.01, 3.02) given g (x, y, z)…

A: We have to use linear approximation to give an estimate for the value of g (0.97, 2.01, 3.02).

Q: Now our goal is to verify the Gradient Theorem – twice! The Gradient Theorem claims that: of - dr =…

A: NOTE:- We are entitled to solve only three sub parts of a question. Concept Used: Line integral of F…

Q: 15. Computo the divergence of the gradient of a function f (x, y,z) 3 x' y' z

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Q: termine the total differential dH of the function H(x xY+z

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Q: sinz Calculate Rz-4|<T z(z-R)² dz using Residue Theorem. |z-4|<n z(z-n)²

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Q: Minimizing the length of the residual vector r = b – Ax is equivalent of mini- mizing the quadratic…

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Q: Find the linearization L(x, y) of the function f(x, y) at the point. Then find an upper bound…

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Q: The gradient of f(x, y) = e³ ª sin(4 · y) at (x, y) = (3, – 2) is defined as followed: V f(x, y) =…

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Q: Show that linear differential operators with constant coefficients obey the commutative law. That…

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Q: . (a) Find the directional derivative of f at (0,1) in the direction of the vector i = i+ 2j. f(x,…

A: Given that: f(x, y)=x2y+y2+yexy

Q: gradient field that corresponds to function (x, y, z) = y2 -x2 +x e=2 - y3 z

A: This is the question from Vector Calculus.

Q: Calculate the frist order differential of the function f(x,y) = x^{4}*y^{2}+2*x-y^{7}+3

A: First order Partial derivatives

Q: (a) Find the directional derivative of f(r, y, z) = 8ry2-+y at the point (1,1, 1) in the direction…

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Q: Find div F and curl F if F(x, y, z) = x'i – 3j + yzk. div F= curl F=

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Q: A line passes through the point Po(xo, Yo) and has gradient m = tan a. Show that a general point of…

A: Gradient, or slope of a line characterizes the direction of the line and is found out by dividing…

Q: F(x, y) = xyi + (x – y)j and C is the triangle from (4, 0) to (-4, 0) to (0, 4) to (4, 0). - (a)…

A: Given a vector field : F→ (x,y)=xyi→+(x-y)j→ We need to calculate the line integral along some given…

Q: Fundamental Theorem: 2x sin t -dt. Find the value of f'(x). t2 +1 Let f(x) = | %3D

A: We can solve this as follows:

Q: Find the number of zeros of f(z) = 2z* – z + 5z – 1 in the annulus 1 < |z| < 2. %3D

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Q: The directional derivative of the fiunction f (x, y, z) = x²z+ xyz + y² in the direction of u = 2i +…

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Q: stlon 5 Use differential to approximate the change in the value of f(x)= ° as e goes from 2 to %3D…

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Q: Find the gradient of f(x.y.z) = (x² +y² +z²)/2 + In (xyz) at the point (2,1,2). DiDi vil/212)…

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Q: the gradient to find the directional derivatve of the function direction of Q. 8. f(x,y) = e' sin x…

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Q: VECTOR DIFFERENTIATION: If R = e^(−t) i + ln(t^2+ 1) j - tant k. Find: (a) dR/dt, (b)…

A: R=e-ti+ln(t2+1) j-tant k

Q: A circular ring of wire of radius ro lies in a plane perpendicular to the x-axis and is centered at…

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Q: Find the derivative of the function at Po in the direction of A. f(x,y,z) = - 3 ex cos (yz),…

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Q: (a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the…

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Q: Find div F and curl F if F(x, Y, z) = x°i – 4j + yzk. div F= curl F=

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Q: ✓(3) Prove Green's theorem for F = Y Z + x²y³j y (1,1) i (1,0)

A: Solution of the given problem is below...

Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A: we have, Fx,y,z=2yi^+3zj^+xk^ and vertices of triangle 3,0,0,0,3,0,0,0,3 ∇×F=ijk∂∂x∂∂y∂∂z2y3zx…

Q: Use a linear approximation to estimate the change in f(x,y, z) = e2" cos(5yz) when moving from the…

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Q: Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C…

A: ∬Df ∇2g dA=∮Cf(∇g).n ds -∬D∇f.∇g dA Add ∬D∇f.∇g dA to both sides, ∬Df ∇2g dA+∬D∇f.∇g dA=∮Cf(∇g).n ds…

Q: (a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the…

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Q: (Continuation of Exercise 31.) If ƒ is continuously differentiable on [0, a] for a > 0, and f(a) =…

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Q: Find div F and curl F if F(x, y, z) = x*i – 11j + yzk. div F= curl F=

A: Vector

Q: QI If f(2) s analy tic function for all 2 Prove tha t dy

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Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A: Answer is in next step. please give a like!!

Q: Find div F and curl F if F(x, y, z) = x°i – 10j + yzk. div F= curl F=

A: To find the divergence and curl of the following vectorF→=x9i-10j+yzk

Q: A differentiable function f(x, y) has the property that f(1, 4) = 5 and f-(1, 4) = 3 and f,(1, 4) =…

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Q: Find the derivative of the function at Po in the direction of A. f(x,y,z) = 3 ex cos (yz),…

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Q: The gradient of f(x,y,z)=ye-x² sin(yz) (-1, 1, ) is given by OA.e-¹(0, -, − 1). - О в. B. (0,л,…

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Q: Find div F and curl F if F(x, y, z) = xºi-7j+yzk. div F= curl F=

A: Given that F(x,y,z)=x6i-7j+yzk. The objective is to find divF and curlF.

Q: Find div F and curl F if F(x,y, z) = x°i – 2j + yzk. div F= curl F=

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Q: 10 Find div F and curl F if F(x, y, z) = xz°i+ 5y"x²j+ 4z?yk. div F= curl F=

A: Solution:

Q: Problem show that the Force F. (Y²Z²_6X2² ) ( + 2x4²²) 22 +(3xy 2² - 6x³2) Kis Conservative, hence…

A: Where C is arbitrary constant. Which is the required potential function.