Consider F and C below. F(x, y) = 8xy² i + 8x²y j C: r(t) = (t + sin(at), t t + cos(-1⁄nt)), (a) Find a function f such that F = Vf. f(x, y) = (b) Use part (a) to evaluate Schi 0 ≤t≤ 1 Vf. dr along the given curve C.

Q: - Let f(x, y) =sin(x y − x − 3y+3). (vi) Which of the surfaces below is a graph of this function?

A: Given .This can be rewritten asOn the xy−plane, , if  gives us hyperbolas on the xy−plane.

Q: Isin , it (z, y) # (0, 0) 2. Determine if a function f : R → R defined by f (r, y) = 0, if (z, y) =…

A:

Q: Determine the function f (0) so that x = a cos0, y = a sin0, z = f (0) should be a plane curve.

A: The given relation is, x=a cosθ, y=a sinθ. To Find: A function f such that z=fθ is a plane curve.

Q: Evaluate y dx + x dy along the curve y x² from (0,0) to (1, 1).

A:

Q: Find a function f such that Vf = 2e²x cos(5y)i - 5e²x sin(5y)j + z³k. f(x, y, z) =

A: Given :-  = 2e2x.cos(5y)i - 5e2xsin(5y) j + z5kNow,

Q: (2) Find the work done by F over the curve C in the direction of increasing t. Explain why the sign…

A: Question: Calculate the work done by  Fx,y,z=zi^+xj^+yk^ over the curve  C : r¯t=sinti^+costj^+tk^ ,…

Q: 2.(2.3) Let f(x, y) = xy(e² + e³) for (x, y) = R². (1) Find fx and fy. (2) Find an equation of the…

A: (1) fx​=yex+yey+xyexfy​=xex+xey+xyey (2)(3e)x+(3e)y−z=4e  Explanation:

Q: u(x, y) = -e sin x - 2e* siny. Find a harmonic conjugate v of u, namely a function v such that the…

A:

Q: Q: Find the condition that the line x cos a +y sin a = p may touch the curve x"y" = a™*".

A: Answer

Q: Find a function f such that Vf = 2e²x cos(5y)i - 5e²* sin(5y)j + z5k. f(x, y, z) =

A: The given gradient function

Q: find dw/ar if w = (x + y + z)?, y = cos(r + s), at the point (r, s)3(1,-1) x = r - s, z = sin(r + s)

A: Given w=(x+y+z)2 x=r-s y=cos(r+s) z=sin(r+s) point (r, s)=(1, -1)   when we differentiate partially…

Q: Let r(t) = . Find r′(t), and then find all values of t for which r(t) is smooth.

A:

Q: please answer fully on paper and make sure its correct please i keep getting different answers

A: