Use Stokes' theorem to calculate the line integral f (e² − y³) dx + (e¹ + x³) dy + e* dz, e² when the curve C has the parameterization r(t) = (cost,sin t,sin 2t), 0 ≤ t ≤ 2¹. hint: The curve is formed by the intersection of the saddle surface z = 2xy and the cylinder x² + y² = 1. Another hint: Here again n dS = N dA. (Ans: 3 ¹/2)

Q: Evaluate the line integral / Vxy° +3 dx +6xy" dy, where C any curve joining (1, -1) to (16, 1).

A:

Q: Find a function y = f(x) that describes the parametric x = -2t + 1, y = 3t², for – o <t < ∞. curve

A:

Q: Evaluate the line integral f y(tan x + x sec² x) dx + x tanæ dy + dz once around the circle x² + y²…

A:

Q: Let Yzo.r parametrise a circle with centre zo and radius r (positive orientation). Evaluate the…

A:

Q: 3. Consider $ -y3 da + a?y dy where curve C is defined by the line segment from (-1,0) to (1,0),…

A:

Q: Find the equation of the tangent plane to the surface y = 2x2 – z2 at the point (0, –1, –1) y+ 4z =…

A: We have to find

Q: Evaluate the line integral / (2x + 4y)dx + (2x – 9y)dy along the curve C : x = 4 cos t, y = 8 sin t…

A: Solution is given below:

Q: Evaluate the line integral, where C is the given curve. Lo (x + yz) dx + 2x dy + xyz dz, C consists…

A: We have to evaluate the following line integral  Where C consists of line segments  to and from  to…

Q: The parametric curve below x=2t3, y=(4/2)t2, Integral to find its length from (0,0) to (16,8)…

A:

Q: Evaluate the line integral F. dr where F = (4 sin x, 3 cos y, cz) and C is the path given by r(t) =…

A:

Q: Evaluate the line integral f, F . dr, where C is the curve given by F(t) = 3 cos ti+ 3 sin t j, 0<t<…

A:

Q: (x = t³ + 7 y = t5 – 3t d²y calculate for a function given parametrically dx?

A:

Q: 5. Consider the saddle surface r = y- 2, find the point where the tangent plane is parallel to x+y+z…

A: If two vectors a1, b1, c1 and a2, b2, c2 are parallel, then it can be written that a1, b1, c1=ka2,…

Q: Evaluate the line integral (5x + 8y)dx + (3x – 5y)dy along the - curve C : x = 10 cos t, y = 20 sin…

A:

Q: F. dr where F = yzi + xzj +xyk and C is the curve x = t, y = 2t², z = –2t² with 0<t < 1.

A: Given F=yzi+xzj+xyk and r=ti+2t2j-2t2k Now the vector field along the curve can be given by…

Q: The parametric curve x(t)=t−2t6 and y(t)=t7+2t2 is given. In which case is the tangent line at the…

A:

Q: The line integral x dy – y dx Jc x2 + y? is independent of path.

A: Given integral is ∫Cxdy-ydxx2+y2.

Q: Obtain the integrating factor of the non-exact DE, (2 + ycosx)dx + xcosxdy = 0. a. secx b. cosx c…

A: Integrating factor

Q: |[-y dx + xy dy]

A: See the attachment

Q: tangent plane and the normal line to the surface

A:

Q: dy in terms of t: Differentiate parametrically to find dx 1 y = +2 t+1

A:

Q: 1. Find the equation of the tangent plane to the surface cosh [exp (x²y)] = Z y? at the point where…

A: We will find out the required equation of tangent plane.