.: Evaluate [G(x, y)dx, [G(x,y)dy and [G(x, y)ds on the indicated curve C. G (x, y) = x³ y² + y² sin 3x + cos 3x, x = 3t, y = t,0 < t<1.

Q: 2- Show that the function is analytic everywhere except at the points z +i += and on the half line…

A: The region where a complex valued function is analytic is actually its domain of definition. A…

Q: 1. Find an equation for the tangent plane to the surface x cos z + yře*z =4 at Po(3, –1, 0)

A:

Q: Find T, N, and k for the space curve r(t) = - 2e cos t)i- (2e sin t) j+ 2k.

A:

Q: Find an equation of the tangent plane to the given surface at the specified point. f(x, y) = tan xy.…

A: From the partial derivative with respect to x and with respect to y

Q: Compute the line integral Sc 2.xy dx + x² dy along the following curves. (a) C1 along the circle x2…

A: In given question that is a conservative vector field mean exact integration in which intrgration…

Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (c) Determine the…

A: Here is the solution of the given problem.  To solve this problem we use simple concept of…

Q: Find T, N, and k for the space curves r(t) = (6 sin 2t)i + (6 cos 2t)j + 5tk

A:

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

A:

Q: If z =t cost, y =tsint, z=t, then x² + y² = t² cos²t+t² sin² t = t² = z², so the curve lies on the…

A: Given x=tcost, y=tsint and z=t and z2=x2+y2 Substitute x=tcost, y=tsint and z=t in z2=x2+y2…

Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A:

Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: Let C be the curve given by r(t) = (t cos 1, 1 sin 1, 1). (a) Show that C lies on the cone x² + y² =…

A: GivenParametric equation is r(t)=&lt;tcost,tsint,t&gt;To show C lies on cone x2+y2=z2.

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…

A:

Q: Given the path C that is the line segment from (-4; 0) to the origin, and the curve y = 2x² from the…

A:

Q: 1. Evaluate xyz dz, where C is the curve 7(t) =t cos(t)î + sin(t)ĵ – cos(t)k, 0 < t <1.

A:

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…

A:

Q: 5) Evaluate the integral (e* - y) dx + (sin y + x)dy using Green's theorem where Cis bounded by y =…

A:

Q: b) Find equations for the tangent plane and normal line at the point po on the given surface.…

A:

Q: 4. Find f such that F = Vf and use it to evaluate SF. dr. (a) F(x, y) = (3 + 2xy²) î+ 2x²y ĵ, C is…

A: (4) To find: f such that F→=∇f and To Evaluate: ∫CF→·dr→. For the following Functions: (a)…

Q: Use Green's theorem to evaluate (2y dx-x dy) around the closed curve C. 27 (2,8) 8. 4. C. -2 (1,-1)

A:

Q: Find the work done by F in moving a particle once counterclockwise around the given curve. F= (4x-…

A:

Q: Find the equation for the Tangent Plane on the given surface coS ax – x²y + e*z+ yz = 4 Po(0, 1, 2)

A:

Q: 2. Find the work done by F in moving a particle once counterclockwise around the given curve C F =…

A:

Q: 4. Let f(x, y) = sin(x²y) + vx + e+y. Find the equation of the tangent plane to the surface z f(x,…

A:

Q: Find T, N, and k for the space curve r(t) = - ti - (a cosh (t/a))j, a > 0.

A: Unit Tangent. Unit Normal. Curvature.

Q: Verify the Green's theorem for the following integral $. sinydx + (x – cosy) dy where curve C is the…

A:

Q: The length of the curve r(t)= [cos(t), sin(t), – 3t]with 0 <t<2n is equal to O A. 2/5 T O B. 2/6 T O…

A:

Q: 2. Let F(x,y) = (cos(x – 2y), –2 cos(x – 2y)). Find curves C and C2 so | that, F. dr = 0 F. dr = 1

A:

Q: The volume of the solid generated by the curve y = e 2, between z = 2 and z = 6 about the a - aris…

A: Solution-

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.

A:

Q: Evaluate |(2y+cos x)dx + (5x+ sin y)dy along the positively oriented curve C. C is the boundary of…

A: We have to evaluate the integral ∫2y+cosxdx+5x+sinydy along the positively oriented curve C. Where C…

Q: y = cos(x), 0 <x<

A: Consider the given surface whose area is required to calculate.

Q: Calculate the integral ∫C(x2+y2)zds, where the curve C is parameterized by r(t)=(sin3t,cos3t,4t),…

A:

Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: Given:           x3−2y2+xy=0,     (1,1)

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: A polar curve is given by the differentiable function r = † (0) for 0 sos 2n. If the line tangent to…

A:

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

A:

Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: Consider the given function, xy+y2-3x-3=0 now differentiate with respect to x,…

Q: %3D a) Find fyxyx for f(x,y) = In(xy) – cos (xy) at (1,t).

A:

Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

A:

Q: Find the length of the following​ three-dimensional curve. r(t)=cos t, sin t, 3t​, for 0≤t≤2

A: Given that rt =cost,sint,3t  for 0≤t≤2 find the length

Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3…

A:

Q: Find equation of (a) the tangent plane to the given surface at the specified point. y = e^x cos (z)…

A:

Q: Consider the curve, y with parameter (, and function f in 3 dimensional Euclidean space with…

A:

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

A: