) Verify Stokes' Theorem for the hemisphere S : x²+y²+z² = 9, z > 0, %3D its bounding circle C: 1? +y = 9, z = 0, and the field F = y - x). %3D

Q: B) Use Green's Theorem to evaluate /F-dr, where C is the circle r² + y? = 4 and F (r, y) = (y – 2…

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Q: c) Use Stokes' theorem to evaluate F. dr where F = z' i+yJ + xk and C is the triangle with vertices…

A: Stokes theorem is used to find the line integral of the vector point function

Q: Use Stokes' Theorem to find the circulation of F = 4yi + 4zj+ 3xk around the triangle obtained by…

A: We have to find ∫CF→·dr→ using Stokes'sTheorem

Q: Find polar coordinates (r,θ) where r < 0 and 0≤θ≤2π of(√3,−1)

A: Given point in the rectangular coordinates is 3,-1. To find the polar coordinates r,θ, where r&lt;0…

Q: Identify the surface for ρ = 4 cos φ

A: The given curve is: ρ=4cosφ Multiplying both the sides by ρ: ρ2=4ρcos φ Now, converting it into…

Q: c) Verify Stokes's Theorem for F = (x²+y²)i-2xyj takes around the rectangle bounded by the lines x…

A: Stoke's law is given as ∫cF→.dr→=∫s∫∇×F.n→ds F=x2+y2i^-2xyj^ dr→=(dxi^+dyj^) So F.dr=x2+y2dx-2xydy…

Q: Sketch the parametric curve of r(t) =.Then, find and draw the tangent vector r'(0)

A: 2cos t, 2sin t,3To Find:a. Sketch the parametric curve.b. Find and draw the tangent vector r'(0).

Q: Evaluate /| cos(x? + y²) dA, where D is the D disk x? + y <I Select one: a. -n O b. A O c. O O d. 2…

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Q: The vector function r(t) (5 – 2 sin t) i+ (3+2 cos t) j+ 2k traces out a circle in 3-space as t…

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Q: Describe the projections of the circle r(t) =〈sin t, 0, 4 + cost〉onto the coordinate planes.

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Q: 8) Given z =f (x.y) =1- (x² +4y2) ,Find : a)The vector function, 10 normal to the surface of f at…

A: This is problem of vector calculus.

Q: Let l be the line y = - (7x + 5) in R21 Find vectors dland b so that v = a + tblis a parametric…

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Q: Use polar coordinates to integrate under z=xy above the first quadrant of the unit circle on the xy…

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Q: Vector F is mathematically defined as F = M x N, where M = p 2p² cos + 2p2 sind while N is a vector…

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Q: The regon, D, is the region enclosed by the parametric curve: r(t) = vector brackets (cos(t), sin(t)…

A: Given that the region D is enclosed by the parametric curve,  rt=cost, sint-cost with 0≤t≤2π Area of…

Q: The vector function r(t) = (1+2 cost) i+3j+(5 – 2 sin t) k | traces out a circle in 3-space as t…

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Q: Find a parametric function for the intersection of r – 3y + 4z = 0 and a? + y² = 9.

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Q: Let F(x, y, z) = (6xyz+4x) i+(3x²z+ze")j+(3x²y-e=v) k, and let C be the portion of the parametric…

A: Given F→x,y,z=6xyz+4xi^+3x2z+ze-yj^+3x2y-e-yk^ And C be the portion of the parametric curve given by…