(6) Use the Divergence Theorem to evaluate dS, where F(x, y, 2) = xyi + (y² + e*=*)j + sin(xy)k and S is the surface of the region E bounded by the parabolic cylinder z = 1 – a? and planes z = 0, y = 0 and y + z = 2, as shown below;

Q: 6. Evaluate the line integral F · dr where F = (x³ —y¹, ex²+z², x² – 16y² z²) and C is the curve x²…

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Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

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Q: Use Green's Theorem to evaluate Se (y? – tan-1 (r)) dr + (3r + sin (y)) dy, where C is the boundary…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 8. Change to cylindrical coordinates and find the divergence and curl of (a) F xi + yj x² + y² -yi +…

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Q: Let u(x,y) satisfy the following equation uxx+(x2+y2-1)uyy=0 Find the region in which the equation…

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Q: The quotient of 33288

A: Given: 33288

Q: f(x,y,z) = x e Yz. a) Find Vf = Type an exact answer, using radicals as needed.) b) Determine the…

A: Follow next step

Q: Use the map G(u, v) = a = 4, b = 8, c = 6. b J D a y=cx uv v+1' v+1 (x + y) dx dy = to compute f (x…

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Q: Which of F(x, y) = (-y, x) and G(x, y) = (y,x) is conservative in R2 and why? If again F(x, y) =…

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Q: (a) Compute directly I = S Ss F-dS, where S is the surface of the sphere x? +y? +z? = 4 and F =…

A: X²+y²+z²=4=2² Therefore radius of sphere is 2.

Q: Calculate line integral F-dr where F(x, y) = (7x + y, 5x + 2y) and C is a rectangle with vertices…

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Q: f(x, y) = x - y“; R is the disk x +y < 1.

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Q: ve = Vti + Vrj + (2t – 1)k is tangent to the surface x+y-z = 1 Show that the curv when t=l

A: Consider the given curve r=ti+tj+2t-1k And the given surface x2+y2-z=1 First, write the curve in…

Q: Let C be the counterclockwise boundary of the region enclosed by the x-axis, the line y = x – 2, and…

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Q: Evaluate the line integral (2x-y+6)dx + (5y + 3x - 6)dy, for which the path C traverses C around a…

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Q: Use Green's theorem to evaluate the line integral r ry dr+r² where C is positively oriented curve…

A: Introduction: The process of determining the area of the region under the curve is known as…

Q: (a) Show that the two surfaces S₁: z = xy and S₂ : z = ³x² - y² intersect perpendicularly at the…

A: As per our guideline, we are supposed to solve only first question. Kindly repost other questions as…

Q: Consider the surface F(x, y, z) = x¹☆ + sin(y¹¹) — 9 = 0. Find the following partial derivatives…

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Q: b) Given F(x, y, z) = (x³ + cosh z) i+(2y³ – 3x²y) j – (x² + 4y²z) k. Use Gauss's theorem to…

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Q: Let C be the simple closed curve which consists of the line segment from (1,1) to (-2,4) followed by…

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Q: Given the line integral | (2x2° + 6y) da + (6x – 2yz) dy + (3a² z² – y²) dz. Let C be the path that…

A: Note that the following integral reduces like this :

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: Evaluate the line integral 2xyz dx + x²z dy + x?y dz where C is an oriented simple curve connecting…

A: Given line integral is,

Q: Solve the differential equation by using variation parameter method y''-6y'+9y=8+e^-9x

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Q: Evaluate the line integral |(3="y + (1- y):) dz + (z² – z2) dy + z(1 – y) dz where C is any curve…

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Q: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y

A: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y