Use Gauss' Divergent Theorem to calculate F. dS where F = (2x + sen (et1)) i+3y²j+ cos(z²)k and S is the boundary of the solid E = {(x, y, z) E R³ | x² + y? < 2 e -1

Q: 8.) What is the distance between an airplane flying at 120 feet above sea level and a turtle…

A: We find the distance between an airplane flying at 120 feet above sea level and a turtle swimming at…

Q: Consider the function 3..2 f(x, y) = cos(rxy) – a°y?. = cos(Txy) – x°y°. a) Find the equation of the…

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Q: 3. Consider the following helix with vector equation 7 (t) = 2 cos (t) 7 (t) 7 + tỉ + 2 sin (a)…

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Q: Evaluate F dr where F(x, y, z)=(x, xy, yz) and C is the curve r (t) = (t, t

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Q: y2 +Cos f (x, y, z) =

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Q: (a) r(u) = e" i+e"j. u € [0, 1]. r(u) = (1- u)i, ue [0. 2].

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Q: 1. Consider the surface defined by z = x2 + e9y ln(x-y). Let f(x,y) =  x2 + e9y ln(x-y). ○ Compute…

A: The given problem is to find the gradient vector of the given surface at point and to find the…

Q: f1(x) = sin(x1) cos(x2), T f2(x, y) = xy, f3(x) = xx¹, € R² x = x, yern XER"

A: Given the following functions                  f1(x)=sin(x1)cos(x2),   x∈ℝ2f2(x, y)=xTy,   x, y∈ℝn…

Q: Show that F dr is independent of its path, given that F(r. y. :) = ( i+ tanrj+, + tan k. 1+:2 %3D…

A: Here we will find out the required value and prove the result.

Q: Show that F. dr is independent of its path, given that F(л, у, 2) — + tan 1+x² 2) i+ tanrj+ k. 1+2²…

A: We will show the required result.

Q: Evaluate the line integral f F dr, where F(x, y, z)= -sin xi - 3 cos yj+xzk and C is given by the…

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Q: Calculate dw/du and ðw/dv using the following functions: w = f(x,y, z) = 3x² – 2xy+4z² - x = x(u, v)…

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Q: 1. Let F(x, y, z) = (2re²*+3», 3e* +3v +2/E, ). 2.x cez²+3y a. Show that F is conservative using its…

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Q: Let F = (2x³y² + 2x, 2x¹y³ + y). Given that f(x, y) = x¹y¹ + ² + y² is a potential function of F.…

A: The given function is F→=2x3y4+2x,2x4y3+y, and its potential function is f(x,y)=12x4y4+x2+12y2. To…

Q: 1. Let F(x, y) = (y, 2x2). Use the circulation form of Green's theorem to calculate the circulation…

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Q: Let F= SF. Use Stokes' Theorem to evaluate Z C is the curve connecting four line segments with…

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Q: Work done by the force F(x,y) = (5x + 4cos(y))i + (3y - 4xsin(y)) acting along the curve y=x4 for…

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Q: Let f(x,y)= e, Compute the cartesian equation of the tangent plane r,y) at the point (2.2).

A: The equation of tangent plane

Q: Consider the one-form w = x dx +y dy + z dz on IR3, and the smooth function o : R → R³ given by $(t)…

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Q: Consider the one-form 1 (-y dx + x dy) w = (x² + y²)ª on U = R² \ {(0,0)} , and the smooth function…

A: Given ω=1x2+y24-ydx+xdyϕr,t=rcost,rsint

Q: Find the Jacobian a(x, y) / a(u, v) for the indicated ch x = e-u sin v, y = e-U cos v a(x, y). a(u,…

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Q: Let C be the circle centered at the origin with radius 1 oriented clockwise, and let F(x, y) = ( −6x…

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Q: Ql: f(x.y)-y cos xy over the rectangle 0sxs , 0 sys1

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Q: Scy dx –x dy; C is the cardioid r=1- cos 0.

A: