B) Use Green's Theorem to evaluate F dr, where C is the triangle with vertices (0,0) (3,0) and (3, 4). F (x, y) = (3y² – tan æ)i+ (5xy + In y)j.

Q: 2. Use Green's Theorem (and only Green's Theorem) to evaluate |(2? + y)dx + (x² – y³)dy, where C is…

A:

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: Use Green Theorem to evaluate C P xydx+e'dy where Cis the path from (0,0) to (1,1) along the graph…

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Q: Use Stoke's Theorem to evaluate Į F· dr where F (x, y, =) = , and C is the triangle with C vertices…

A: given Fx,y,z=z2,y2,xy ⇒Fx,y,z=z2i^+y2j^+xyk^ and a traingle C with vertices 1,0,0 ,0,1,0 ,0,0,2 to…

Q: Evaluate F dr where F(r, y, z) = (2.75yz cos(ryz), 2.75rz cos(ryz), 2.75xy cos(ryz)) and C is %3D 1…

A: Here we will find out the required value.

Q: Let F (r, y, z) = sin y i+r cos yj- sin z k.

A: Given,        F(x, y, z)=sinyi+xcosyj-sinzk.

Q: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

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Q: Sketch the strophoid shown below. r= sec(0)- 2 cos(8), -<0 < x/2 1.0 1.0 0.5 0.5 -10 -0.5 0.5 1.0…

A: A strophoid is a curve generated from a given curve C and points A (the fixed point) and O (the…

Q: 3. Use the theorem in Sec. 44 to show that La (z-zo)"-1 dz = 0 (n = ±1, 12,...) when Co is any…

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Q: Use Stoke's Theorem to evaluate $. F · dR, where F (x, y, z) = (z²,xy, xz) and C is the triangle in…

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Q: Sketch the triangle defined by A and compute LA fdA for A = {(x, y) E R²;0< x <a and y <x} and ƒ: A…

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Q: 35. f(x, y) = x+2y2 -x; R is the disk x2+ y? < 4.

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Q: Evaluate O F.dT where +(3y %3D y + 1 and C is the triangle with vertices: (0,0), (6,0), (3, 5).

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Q: Find the first and second fundamental form of x(u,v)=(e(u-v)/2 cos(u+v)/2, e(u-v)/2…

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Q: = Let Co be the curve formed by the intersection of the surfaces x₁ = 2x² and X3 3x1x2, and let C be…

A:

Q: 39. Show that the graph of r =t sin ti + t cos tj + t'k lies on the paraboloid z = x² + y².

A: Here we compare r as its basic equation to prove this.

Q: f (lnx + y) dx − a² dy, where & is the rectangle with vertices (1, 1), 1. Use Green's Theorem to…

A: Let I =∮(ln x+y)dx-x2 dy, where the region is C which is a rectangle with vertices…

Q: Let r(t) = (sin(t), cos(t), 5 sin(t) + 4 cos(2t)). Find the projection of r(t) onto the xz-plane for…

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Q: Use Green's Theorem to evaluate | (v? – tan-1 x²)dx + (3xy + 4x – In (y2 + 4))dy where C is the…

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Q: Use Stokes's Theorem to evaluate F(x y₁ z) = x² +2 - xyzk F. dr. In this case, C is oriented…

A: The solution is given below

Q: Is h: (-1,1) → R, h(x) = tan[(t/2)x] homeomorphism? Please search. 14:42 / %3D a

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Q: 14. Under the paraboloid z=x² +y2 and above the disk x +y² <9 for (x,y) in the first quadrant.

A: we have to find the volume

Q: Show that the function u = p(ry) + JFÿ¢(y/x) provides the quation „Pu = 0. dy

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Q: Use Green's Theorem to evaluate the line integra Icon (y − x)dx + (2x − y)dy - - for the given path.…

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Q: Sketch the bifurcation diagram for y = (A– 4t)y (where A is a parameter, as usual). 2

A: To find- Sketch the bifurcation diagram for y' = λ - 4t2y (where λ is a parameter, as usual)

Q: Given F=. By using Gauss's theorem, find f F ds for a paraboloid x = 4-y² - 2² with x ≥ 0 as shown…

A: We are given  F=-3x+1+z3,y+1+x3,6z+xy So, the divergence of F is div…

Q: Let C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, -1) to (0, 0), C₂ is the line segment…

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Q: 5. Verify Green's Theorem in the plane for F = (xy + y²)î + (x – y)f and the curve C is as shown in…

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Q: Show that U (x, y) = e"cos v,V(x, y) = e"sin v are conjugate harmonic of each other if f = u + iv is…

A: Solution given below

Q: Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y)…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Exercise 10.2. Given = sin u and u = 3x+ 1, find = f'(g(x))g'(x). dy %3D %3D %3D dx

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Q: Let T(r,s,t) = (r cos(s) sin(t), r sin(s) sin(t), r cos(t)) Find the Jacobian of T at (3, π/4, π/3)

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Q: Let Ci be the rectangle with vertices (0,0), (2,0), (0,1), and (2,1) oriented counterclockwise, and…

A: Let C₁ be the rectangle with vertices (0,0). (2,0), (0, 1), and (2,1) oriented counterclockwise, and…

Q: 1. Use a change of variables to evaluate || ソーx dA where R is the trapezoid with vertices (1,0,…

A:

Q: Evaluate F. dr where C is the line segment from (6,,1) to (36, 1, and F = (sin (yz) + 1)i + (zzcos…

A: Whenever a Vector is given and we need to find line integral with two points mentioned, we first…

Q: |F - dr where C is the line segment from (6, 5,1) to (45, 1, ) and F = (sin(yz)+1)i + (zzcos(yz) +…

A: To evaluate:

Q: use Stoles's Theorem to evaluate F ds where Ě= and C is the orde with rading 5 centered C2,0,0…

A: According to Stokes' law:                                                     ∮JF·dr=∬S(Curl F·N)dS…

Q: Express x = In(3t), y = 5 – t in the form y = f(x) by eliminating the parameter. %3D y =

A: Given that x=ln(3t) and y= 5-t we need to write y=f(x) by eliminating the parameter  Now , x= ln(3t)…

Q: 2. If o = x²yz³ and F(x,y,z) = 3yzi + x²j + (x cos y)k, find: a) grad o %3| b) div F c) curl F

A: Given:  φ=x2yz3F→x, y, z=3yzi→+x2j→+x cos yk→ To Find:  a grad φb div F→c curl F→

Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

A:

Q: B) Use Green's Theorem to evaluate F-dr, where C is the triangle with vertices (0, 0) (3,0) and (3,…

A: Explained below

Q: Compute the Jacobian of the mapping at the given point. O(r, 0) = (8r cos(0), 8r sin(0)), (r, 0) =…

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Q: Show that the graph of ř = (t sin t, t cos t, t²) lies on the paraboloid z = x? + y?.

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Q: Use Green's Theorem to evaluate x' dx + xy dy where C is the triangle with vertices (0,0), (1,4),…

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Q: 1. Use a change of variables to evaluate [[ y- dA where R is the trapezoid with vertices (1,0,…

A:

Q: 7. Let F = (2x sin(y)e", x² cos(y)e*, z² sin(y)e*). Find along the path c:T = (sin(t), t, cos(t))…

A: F=2xsinyez, x2cosyez, x2sinyez curlF=ijk∂∂x∂∂y∂∂z2xysinyezx2cosyezx2sinyez⇒curlF=0 So, F is a…

Q: (b) Given  = (x²y)î + (xz – z²)f – (xyz)k, determine the following at point (1, 1, 2): (i) V- Â.…

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Q: 3.use Greens theorem s (1+x5 dx + 2xy dy ,where C is triangle With vertices (0,0),(1,0),(1,3)…

A: Apply greens theorem

Q: 12. Let H be the hyperboloid where x² – y² + z² = 6. Find an equation for the tangent plane to H at…

A: As you asked only solution for Q12 so, I answered here only Q12