Evaluate F. dr where F = (x – 3y)i + (y – 2r)j and C is the closed curve in the xy plane, x = 2 cos t, y = 3 sint from t =0 to t= 2T7. Ans. 67, if C is traversed in the positive (counterclockwise) direction.

Q: 3) Consider the curve C in R³ given by: r(t) = (t + In t, t – In t, 4t'/²) con 1<t< 2 An expression…

A: Formula: If r(t)=x(t), y(t), z(t), a≤t≤b is parametrization of a curve C then the line integral…

Q: 2. Let y be the curve in R° given by t (t, 2 cos t, 2 sin t) for 0 < t< 47. Compute y? ds

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Q: (b) Consider the curve C with parametirzation 7(t) = (5 cost, 5 sin t, 500), where 0 <t< 27. Let F =…

A: We have,

Q: Find the gardientoff at the indicatedpoint. frx,y)=D5sin x+Cos3y ;

A: Given: f(x,y)=5sinx2+cos3yπ2,0

Q: 2. Let R(t) = sin 4t î – bcos"(21) j + 2t k. Find the moving %3D trihedral of the curve of R(t) at…

A: Given

Q: Evaluate the line integral along the curve C. (xz+2) ds, C is the curve r(t) = (-6-2t) i + tj- 2tk,…

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Q: Determine the exact area between the x-axis and the curve defined parametrically by ¤(t) = ;t2 1…

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Q: Calculate F.dr, where C is the curve given by ti + 123+ tk, with t = [0, 1] and F(x, y, z) = 4xei +…

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Q: Let C be the curve defined by T(t) = (2 + 2 sin(t), 3 – 2 cos(t)) for 0 < t < 4x=²) dx + (2xyev" +…

A: Given the integral is ∫Cy2exy2+4x-2dx+2xyexy2+2dy. Rewrite the integral as ∫CF·→dr→:…

Q: Evaluate Fdr where F (x, y) = ( arctan y, 1+y and the curve C is given by C : x = 2 cos t, y = 2 sin…

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Q: Find the moving trihedral of the curve R(t) = (e^t)sin(t)i − (e^t)cos(t)j + (e^t)k at any t in the…

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Q: Com pute I = (22132°y+ 4xy)4x+(x'+2x²-y²)dytzk + 4xyLdx + (x°t2x² -y?) dy+zdz along The curve C C4U…

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Q: 6. Find F' and dF[a; h] where: In ¤1 (a) F(x1,x2) and hence show that if v is tangent to the curve…

A: (a.) Given: Fx1,x2=lnx1x2e2x1x22cosx1 To find: F' and dFa;h To show: If v is tangent to the curve…

Q: Use Green's Theorem to evaluate F. · dr, where F = (Va + 4y, 2x + and C consists of the arc of the…

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Q: Evaluate ∫c (y+sin x) dx + (z2 + cos y ) dy +x3 dz where C is the curve r(t) = (sin t, cos t, sin…

A: ∫C y+sin x dx + z2 + cos y dy + x3 dz = ∫C y+sin x i + z2 + cos y j + x3 k (dx i + dy j + dz k)…

Q: Consider the following. x = sin(2t), y = -cos(2t), z = 8t; (0, 1, 47) Find the equation of the…

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Q: Find F. dr for F = 4yi + 2x j - 2z k if C is the helix x = cos t, y = sin t, z = t, for 0 <t < 4n.

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Q: Evaluate the integral | (3x – 2 y + :)ds where C is the curve, C, and C, as shown. 3- y² + z² 2. 3…

A: Here, Curve C is equivalent to c1+c2. The point of intersections of c1 and c2 is at 0,1,0.

Q: Let C be the curve.

A:

Q: Consider a new curve Č parametrized by r(t) = (cos t esint, sin t- cos t), 0 <t <I but with F still…

A:

Q: 1. Evaluate f(xy+2²)ds where Cis an arc of the helix C x = cost, y=sint, z = t which joins the…

A: Note:- As per our guidelines, we can answer the first part of this problem as exactly one is not…

Q: Given the force field F = (2xze*z – 4, cos³z, x²e**z – 3ycos²z sin z) Solve for the work done by F…

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Q: Consider curve C at R parameterized by: r(t) = cos(2t) î + tf+2 sin(2t)k, with 0 stsT An expression…

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Q: Find the work done by F = xyi+yj - yzk over the curve r(t) = ti + t²j + tk, 0≤t≤1

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Q: Find the work done by force F(x, y) = 4yi + 7xj + (x + y)k in moving an object along curve r(t) =…

A: To find the workdone along the given path

Q: Let C be the curve parameterized as follows F(t) := (9 cos(t), 9 sin(t), 12t), 0<ts 10 and let f(x,…

A:

Q: Find an equation of the plane tangent to the following surface at the given point. V3 A,1, sin (xyz)…

A:

Q: Suppose C is the curve r(t) = (6t,t°), for 0sts2, and F = (3x,3y). Evaluate F•Tds using the…

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Q: Let z = g(x, y) = f(3 cos(cy), y + ey) provided that f(3,5) = 8, fi(3,5) = 2, f2(3, 5) = 3. i) Find…

A: Chain rule of double derivative: Let us consider a two variable function z=f(u,v) where u=p(x,y) and…

Q: Consider the curve C in RS parameterized by r(t) = sin(2t) î + cos(t) ĵ+t k, con 0 <t < 2n An…

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Q: Evaluate (y + 3 sin(x)) dx + (z2 + 9 cos(y)) dy + x³ dz where C is the curve r(t) = (sin(t), cos(t),…

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Q: Evaluate (y + 3 sin(x)) dx + (z2 + 4 cos(y)) dy + x³ dz where C is the curve r(t) = (sin(t), cos(t),…

A: Given: ∫Cy+3sin xdx+z2+4cos ydy+x3dz to evaluate. The parametric equation is rt=sint, cost, sin 2t,…

Q: Let C denote the curve parametrized by a = cos(Tt), y = 3t, z= sin(nt), 0 <t < 1. Then S a2 + y? +…

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Q: Find the work done by F over the curve in the direction of increasing t. F= 11zi + 11xj + 11yk, r(t)…

A: We have to find the work done by F over the curve

Q: 13. Let F (x, y) = (1(²y²)x, af(x²y? , a(y²) Y) and suppose that C is a smooth curve parametrized…

A:

Q: 18. Integrate f(x, y, z) = – Vx² + z² over the circle r(t) = (a cos t)j + (a sin t)k, 0 <t< 2m.

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Q: Use Green's Theorem to evaluate | F. dr, where F = (Va + 5y, 2x + Vỹ ) and C consists of the arc of…

A:

Q: Use Green's Theorem to evaluate / F · dĩ, where F = (VE + 3y, 2x + vũ) C consists of the arc of the…

A:

Q: Find the work done by F over the curve in the direction of increasing t. F=xyi + 8j + 6xk; C: r(t) =…

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Q: 2. Let R(t) = }sin 4t î – bcos“(21) j + 2t Ř. Find the moving %3D trihedral of the curve of R(t) at…

A:

Q: Find an equation of the plane tangent to the following surface at the given point. z = tan (xy);…

A:

Q: 3. If Ø = 2xyz?,F = xyî – zĵ + x²k and C is the %3D curve x = t2, y = 2t, z = t³from t = 0 to t = %D…

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Q: Compute the length of the curve r(t) = 8t cos(t), 8t sin(t), 2t over 0 st < 2n. Hint: V? + a2 dt = +…

A: Given

Q: 2. Let y be the curve in R3 given by t (t, 2 cos t, 2 sin t) for 0 <t < 47. Compute ds

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Q: Find the work done by F over the curve C in the direction of increasing t,where F=-5 yi+5xj+8z°k and…

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Q: F(r, y) = (2xy³ +3)i+ (3x²y² + 2e²»)j is conservative by finding a potential function f for F, and…

A:

Q: Find the length L of the curve R(t) = 4 cos(5t) i – 4sin(5t) j + tk over the interval [3, 6]. L =…

A: Given:

Q: Let C be the part of the curve x? + y² = 1 with y 2 0, oriented counter- clockwise. Evaluate F. dr,…

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Q: Find F. dř for F = 5yi – (sin y)j on the curve counterclockwise around the unit circle C starting at…

A: