Question 1 Let F(x, y, z) = (x¹/³)i+y√1+2zj+e³k and consider the path c(t) = t³i+tj+t²k for 0 ≤ t ≤ 2. (a) Calculate F(c(t)) · c'(t). (b) Evaluate the line integral J F.ds.

Q: Let F(x, y) = x2 + x² + y2J- (a) Show that F. dr +| F. dr if C and C2 are the semicircular paths…

A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

Q: 21 Let F(x, y) = (3x + 2y)i + (2x - y)j. Evaluate F. dr where C is the curve X y from (0, 0) to (1,…

A: Here we calculate the integral along the curve x = y2 from (0,0) to (1,1)

Q: (1 point) Evaluate the line integral y dx + x dy where C is the parameterized path x 1 x = t², y =…

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Q: Evaluate ʃC F dr where C is the parabola y = x2 from (0, 0) to (2, 4) and F (x, y) =< −ysinx,…

A: Given a curve C is a parabola from (0, 0) to (2, 4)

Q: Let  f(x, y) = sin(15x2 + 3y2) Calculate the quadratic approximation of f at the point (0,0)

A: Given that,  fx, y=sin15x2+3y2 The quadratic approximation of the function fx, y at the point x0, y0…

Q: F(x, y) = (2x - y, x + 3y), and let C be the path along the parabola y = 2x² from (0, 0) to (2,8).…

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Q: 13. Find an equation for the tangent plane of the graph of f at the point (xo, yo, f(xo, yo)) for:…

A: Note:-Since you asked multiple questions which are not interlinked to each other, according to our…

Q: b) Evaluate S(2y + 4z)dx + (2æ + 3z)dy + (4x + 3y)dz, where C is the curve given by r(t) = for 0 <t…

A: given ∫C2y+4zdx+2x+3zdy+4x+3ydz where C is the curve given by rt=t3,2sinπt2,3cosπt2 for 0≤t≤1…

Q: 1. Consider the vector-valued function F(t) = sin(2t) t (2, 1, -2), a) Find the domain of F. b) Show…

A: Disclaimer: There are two questions here and as per Bartleby policy, I am solving only the first…

Q: Evaluate √(2² + yz sin(xyz))dx+(y²+xz sin(xyz))dy+(x+xysin(xyz))dz where C is the curve following…

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Q: Let F(x, y) = x²e*i+ 60x²y² j, and let C1 denote the curve consisting of the line segment from (0,0)…

A: Given, F(x,y)=x2ex,60x2y2 The curve C1 consists of the line segment Ca from 0,0 to 1,1, followed by…

Q: Let P be the curve that follows the helix r = from (1,0,0) to ( 1/2,√3/2 , π/3 ), and then returns…

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Q: Suppose F(x, y) = x² + y² and C is the line segment from point A = (1, -1) to B = (3,-5). (a) Find a…

A: Given          Let       (a)   The equation of the line segment from A(1, -1) to B(3, -5) is,,…

Q: f (x, y) is a differentiable function with Vf(4,3) = 7i+ 5j. 2. Suppose w = (a) Find the directional…

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Q: Find the tangential and normal components.  r(t)=(t+1)i+2tj+t^ 2 × k     ,   t = 1

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

Q: Compute F · dr where F(x, y) = and C is the path traced by the line segment starting at (0, −1) and…

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Q: Sketch the curve r(t)=f°i+tj and indicate the orientation. Find r'(t) and sketch r(1) & r°(1).

A: First we will plot the given parameters.

Q: 6.(6 pts) Evaluate the line integral / (x+ y + 2) ds along the curve C given by r(t) = (sin t, cos…

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Q: F(x, y) = x² i + y² j C is the arc of the parabola y = 4x² from (-1, 4) to (0, 0) %3D (a) Find a…

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Q: Consider the tangent line to the curve r(t) = t2i −(1/(t + 1))j +(4 − t2)k at the point P(4, 1, 0).…

A: Given the tangent line to the curve r→t=t2i^−1t+1j^+4−t2k^  at the point P4,1,0. We have to…

Q: 2(x + y)i + sin(y) 3. Find the line integral of Faround the perimeter of the rectangle with corners…

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Q: Let F = e'i+(xe' -e')j- ye'k and cbe a smooth curve defined by Then the value of the line integral…

A: The formula is  ∫F.dr=∫F(r(t)).r'(t)dt Here…

Q: Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0).…

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Q: y The circle x² + y² revolved about the line 1. Phat is the Volume (1 2 = 31

A: We have to find the volume of a circle x^2 +y^2=36 revolved by the line y=4.

Q: Let C be the simple closed curve which consists of the line segment from (1,1) to (-2,4) followed by…

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Q: Find an equation (x(t), y(t), z(t)) of the line that is the intersection of tangent planes at (1,…

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Q: . Evaluate SF.dr: ▪ F = (x+z)i +zj+yk. C is the line from (2,4,4) to (1,5,2).

A: Since all subparts are different and not interlinked so as per our guideline we are supposed to…