One of the following vector functions is a gradient Select one: O (xy? + 2y?)7+ (2y³ – x²y+ 2x)j O e'cos yi+e' sin yj O (xe" +x?)7+(ve- 2y) O (2ry +3 - y sin.x)i + (x? + 2y + 1+ cos x)

Q: Find the gradient of U = 2p(z² + 1) cos p. 2 cos ф (2?)ар + 2 sin ф (z?)aф + 4pz cos ф a, 2 cos ф (z…

A: The given function is, U=2ρ(z2+1)cosϕ. Obtain the value of the gradient.…

Q: Suppose that f(r, y) ry. The directional derivative of f(x, y) in the directional (1, - 1) and at…

A: directional derivative f(x,y) = adf/dx +bdf/dy (a,b) = (1,-1) df/dx = y df/dy = x directional…

Q: 4. Find the directional derivative of f in the direction of vector v at point (1,3) such that v =…

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Q: Find work done by the gradient of f(x, y, z) = zexy- 2y2/z  along the path from (1, 2, 4) to (-1, 0,…

A: By Fundamental theorem line integral,we have Where C is curve joining to

Q: Write down the the definition (in terms of limits) of the directional derivative and use it to…

A: We have to write down the definition of the directional derivative and also find the directional…

Q: 2. Suppose that f (1,2)= 3 and that the directional derivative of fat point (1,2) in the direction…

A: Given that f1,2=3 and the directional derivative of f at point (1,2) in the direction of the vector…

Q: .    Find work done by the gradient of f(x, y, z) = zexy- 2y2/z  along the path from (1, 2, 4) to…

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Q: Find the directional derivative of f(x, y, z) = zy + xª at the point (2, 1, 3) in the direction of a…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: Choose the gradient of the function f defined by f(x,v,z) = cos(xy +9xz + 13yz). O sin(xy O -cos(…

A: Take gradient and solve

Q: Let F(x,y,z)=3x2+3y2+4z2. For which unit vector 0 will the directional derivative D0F at  (12,12,12)…

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Q: 4. Show that the covariant derivative of the metric tensor vanishes: V,gjk = V,9k = 0. %3D

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Q: 2. Suppose that f (1,2) = 3 and that the directional derivative of ƒat point (1,2) in the direction…

A:

Q: Suppose that f (1,2) = 3 and that the directional derivative of f at point (1,2) in the direction of…

A: Given : f1,2=3 and the directional derivative of f at 1,2 in the direction of the vector i+j is…

Q: Find the increase rate of the scalar V at point P from origin to the point P(2,6; 9,8 ; 9,5) where…

A: According to question given that The origin to the point P(2,6 ;9,8 ; 9,5) where V=3, 3xy+ 2, 6xyz

Q: : Find the derivative of the function at Po in the direction of the vector AC. f(x, y, z) = x² + 2y²…

A: To find the directional derivative of the given function.

Q: Example 4 Find the directional derivative off(x. y. z) = xy -v+at the point (1,-2,0) in the…

A: If f is differentiable at x,y,z  , then f has a directional derivative in the direction of any…

Q: 2. Find the gradient vector at the indicated point f(x, у, 2) 3 ху — In (2),. (2, -2,2)

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Q: The divergence of the vector G=i+siny j + 3k is :- -8y x3 + siny -8y x3 + cosy Option 1 Option 3…

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Q: The gradient vector of the function f(x, y, z) = ln(yz) - x² is: Select one: ○ (−2x,, 1) O None of…

A: To find the gradient vector of the functionf(x, y, z) = ln(yx)-x2

Q: Prove that the vector given by (sin ysinh x + cos ycosh x) i + (cos y cosh x- sin ysinh x) j is a…

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Q: The gradient vector of the function f(x, y) = cos(4x - 6y) is (4 sin(4x - 6y), -6 sin(4x - 6y)).…

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Q: Choose the gradient of the function f defined by f(x, y, z)= In[1 + (17xy + xz +9yz)²].…

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Q: Suppose ƒ is differentiable at (9, 9), ∇ƒ(9, 9) = ⟨3, 1⟩, and w = (1, -1). Compute the directional…

A: Given that ∇ƒ(9, 9) = ⟨3, 1⟩. Note that w = (1, -1) is not a unit vector. Its unit vector along same…

Q: The gradient vector for the surface xy + xz + yz = 1 at the point (1, 1,0) is (1, 1,0). Select one:…

A:

Q: The gradient vector of the function f(x, y, z) = ln(yz) + x² is: Select one: (2x, /, // ) (2x, -, -…

A: gradf =∇f =i∂∂x+j∂∂y+k∂∂zf .

Q: The gradient vector of the function f(x, y) = sin(4x - 6y) is (4 cos(4x - 6y), 6 cos(4x - 6y)).…

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Q: The gradient vector of the function f(x, y, z) = ln(yz) + x² is: Select one: ○ (2x,-1,-1/2) (2x₂ O…

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Q: Calculate the directional derivative of f(x,y)= In( Vx +y ) at point (1,1) in the %3D direction of v…

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Q: Suppose that f(I,2) =3 and that the directional derivative of fat point (1,2) in the direction of…

A: Thr given problem is to estimate the value of f(1.1,2.1) with given values f(1,2)=3 and directional…

Q: Find First and second derivatives of Funchion . tm - (e-a -a)(e-a+a) Where a is parameber

A: To find a first and second derivative by first simplify the equation and finding it's value at x=0

Q: 2. Prove that the directional derivative of f(x, y, 2) = ry sin z at (1,2, ) in the direction v = i+…

A: Given:        fx,y,z=xy sinz         x,y,z=1,2,π2    direction v=i+2j+2k   Differentiate the…

Q: 3. At point (1,0,3) on the surface f(x,y) = 2x^2 + 3 y^2+ z^2-11, the directional derivative in the…

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Q: Find the two-unit vectors that are parallel to the tangent line to the curve y =x2sin(x) at the…

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Q: Consider the level curves for F (x, y) = x² – y². y (a) How do we sketch in the gradient vectors?…

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Q: Find the directional derivative of f(x, y, z) = xy²z³ at P(3, 1, 1) in the direction of Q(0, -5, 7).…

A: Given function: fx,y,z=xy2z3

Q: If f(x,y) = ln(xy) + x^2y^2, how do I calculate the second directional derivative of f in the…

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Q: Find the directional derivative of the function at the given point in the direction of the vector v.…

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Q: Choose the gradient of the function f defined by f(x, y, z)= In[1 + (7xy + 13xz + 17yz)²]. O…

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Q: The directional derivative of f(x, y, z) = xyz at point (-1, 1, 3) in the direction of vector î - 2ĵ…

A: Remark: We know that the directional gives the rate of change of a function at a specific point in…

Q: What is the gradient of f(x, y, z) = x³z2 + y³z +125z - 1 at P(5,5,-1) ?

A:

Q: 4. Show that the covariant derivative of the metric tensor vanishes: V,gjk = V,gk = 0.

A:

Q: Find the gradient of the function and the maximum value of the directional derivative at the point…

A: The solution is given as

Q: The gradient vector of the function f(x, y) = 2+ Ty at the point (1, 1) is %3D Select one: O True O…

A: The gradient vector of function f is given by: ∇f=∂f∂x,∂f∂y Here, ∂f∂x=12xyy=y2xy∂f∂x=12xyx=x2xy

Q: Find the gradient of the function at the given point. In(x2 - у) - 3, (2, э) Vz(2, 3) = %3D

A:

Q: Find the directional derivative of f (x, y, z) = xy²z³ at P(2,1,1) in the direction of Q(0,–3,5)

A: Solution: The objective is to find the directional derivative of the given function

Q: Find the directional derivative of f(x, y, z) = xy²z³ at P(4, 1, 1) in the direction of Q(0, –7, 9).…

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Q: Calculate the gradient of the function f(x, y) = cos (4z² – 4y) O a.-8z sin (4x2 4y)i + 4 sin (4x2 –…

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Q: The gradient vector of g(x, y) = at (√√2,-1) is Vg(√2, -1) is equal to Select one: O True O False -

A: Given: The gradient vector of g(x,y)=x22-y22 at (2,1) is ∇g(2,1) is equal to <2,1>

Q: What is the directional derivative of f(x,y,z) = ln(x²+2y²+ 4z²) at the point P(-1, 2, 4) in the…

A: The given function is; f(x,y,z)=ln(x2+2y2+4z2) Point P(-1,2,4) u=-3i/13 -4j/13 -12k/13.

Q: Find the work done by the gradient of ƒ(x, y) = (x + y)2 counterclockwise around the circle x2 + y2…

A: