Consider the vector field F(x, y, z) = (3z + 4y)i + (3z + 4x)j + (3y + 3x)k a) Find a function f such that F = Vf and ƒ(0, 0, 0) = 0. f(x, y, z) = b) Suppose C is any curve from (0, 0, 0) to (1, 1, 1). Use part a) to compute the line integral SF · dr.

Q: Define two vector functions F(t) = 8 sin(t)i + 6 cos(t)j + (t – 2)k ü(t) = 6 sin(t)i + 8cos(t)j + (t…

A: We have to find the dot product of r(t) and u(t)

Q: c) Show that the vector field W(x, y, z) = (2yz+ xz – x², –2xz – yz + y?, y² – x² + z²) - - is a…

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Q: 3. Calculate the directional derivative of the function f : R³ → R, defined by f(x, y, z) = y²+xz at…

A: Step 1: Step 2: Step 3: Step 4:

Q: Find r'(t), r(t), and r'(to) for the given value of to r(t) = (1+t)i + ³j, to = 1 (No Response)…

A: We have to find the values of r't, rt0, r't0 where rt=1+ti^+t3j^, t0=1 and sketch the curve…

Q: Find the directional derative of f at the point P0(1,1) in the direction of the vector A = 3i - 2j…

A: Direction derivatives at (1,1) is given in direction of A = 3i - 2j.

Q: Find the domain of r(t) and the interval(s) on which the vector- valued function is continuous. r…

A: We need to find the domain and the interval on which the vector-valued function is continuous.

Q: Consider the force field F(x, y) = (x, y). a) Sketch the vectors F(1,1), F(-2,1) and F(2,0) directly…

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Q: Let f(t) be a function with the following graph. Let F(x)= S", f(t) dt where -4 < x < .. (a)…

A: Hi! You have posted multiple questions. We are answering only the first two parts. If you need an…

Q: Define two vector functions 7(t) = 2 sin(t)i + 8 cos(t)j + t°k ü(t) = 8 sin(t)? + 2 cos(t)j + (t? –…

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Q: F(t) = (Cos(Tt), - 1), G(1) = (1, 1, – 1), G'(1) = (2, 3, 2) , Ĝ"(1) = (0, 1, 0) . (F.G) 1) (Gxở)…

A: Given :- F →t =  cos πt , e2t-1 , t2 - 1  , G →1 =  1 , 1 , -1G' →1 =  2, 3, 2G'' →1 =  0 , 1 , 0…

Q: One of the following vector functions is a gradient Select one: O (xy? + 2y?)7+ (2y³ – x²y+ 2x)j O…

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Q: Let f(x,y,z) = x2 y−xy2+z3, P(1,1,2), u is the vector from P to Q(1, −2, −2). Determine the maximum…

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Q: Let f(x, y, z) = x²y - xy² + z³, P(1,1,2), u is the vector from P to Q(1, -2,-2). Determine the…

A: The given problem is to determine the maximum and minimum directional derivative of the given…

Q: Find r'(t), r(t), and r'(t) for the given value of to. Then sketch the space curve represented by…

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Q: Compute fa(P) where is a unit vector in the direction of . a) f(x, y) = x² + 2xy + 3y², P = (2,1), v…

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Q: For the vector-valued function r(t) = t³i + (2t² + 1)j + (3t − 6)k find the following and express…

A: According to bartleby guidelines we supposed to do the first three subparts of multiple subparts…

Q: Consider the vectorial V=2+ŷ+ 2 z . . function=z²+x²y + y²2 and the gradient operator Please…

A: Given that the vectorial function ψ→=z2 x^+x2 y^+y2 z^ The gradient operator ∇→=x^.∂∂x+y^.∂∂y+z^.∂∂z…

Q: (a) Find the arc length function for the curve y = In(sin(x)), 0 < x < π, with starting point (7,0)…

A: Here, the given curve is y=ln sinx with starting point π2,0. To Find: Arc length .

Q: Compute the vector assigned to the pointP = (−3, 5) by the vectorfield:(a) F(x, y) = (xy,y − x)(b)…

A: To Determine : (a) F(x, y) = (xy,y − x)(b) F(x, y) = (4, 8)(c) F(x, y) = 3x+y , log2(x + y) Given:…

Q: Consider the function f(x,y) = ex cos y + e³ sinx and the points P(1,0) and Q(-3,3). Find Duf at the…

A: The given function is fx,y=excosy+eysinx and the points are P(1,0), Q(-3,3). The objective is to…

Q: Find a potential function of the vector field. F(x,y) = (3x²y + 1, x³+2y)

A: we have to find potential function.

Q: Find the domain of the vector functions, r(t), listed below. You may use "-INF" for -∞o and use…

A: According to bartleby guidelines we supposed to do the first one of multiple questions kindly repost…

Q: Prove that (Y - Y1)'(Y - Y1)= (-Y1)'(- Y1) + (Y-P)'(Y-P) where Y = n 1 Y and 1 is an n-dimensional…

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Q: If the following vector function are equal to: A(t) = 5*t i + 2*t^2 j B(t) = 3*t^2 + 2 j Then…

A: The given problem is to find the derivative of the dot product of the two vector functions. Given…

Q: Calculate vector field F = (2-z²+2-x²)i + (2 · y³ + 4·x)j + (2x²z³+2)k a flux from a box with the…

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Q: Assume that an object is moving along a parametric curve and the three vector function T (t), N(t) ,…

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Q: Find the curl of the vector field F(x,y,z)=x^2yi-2y^3zj-3zk.   A. curl F = 2y3i + x2k          B.…

A: Option (c) correct  For detail...go through the solution

Q: The position of a particle is determined by the vector-valued function r(t) = (1 - 12,3t, 1³). Find…

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Q: f(x,y) = 2x3+4y3-3x2-120x-192y+5 Is f(x,y) a curve, vector field, or surface?

A: To check whether it is a curve, vector or surface. f(x,y) = 2x3+4y3-3x2-120x-192y+5

Q: Find a vector-valued function whose graph is the plane x+y+z= 12, O r(u, v)=-ui-vj+(12-u-v)k O r(u,…

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Q: Let f,g be scalar functions, ủ be a vector function, and assuming sufficient differentiability. Then…

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Q: For the vector field (x, y, z) evaluate the line integral along the curve made of three adjacent…

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