Use Stokes' Theorem to evaluate F. dr. z = 4 – 22 – y? F(x, y, z) = 2zi + 10xj+ 3yk where C is the boundary of the paraboloid shown in the figure to the right. NOTE: Enter the exact answer. R C 2? + y? = 4 F dr ||

Q: Use Green's theorem to calculate §. -y² dx + xy dy where C is the vertex triangle (1,3), (4,0) and…

A: Consider the given integral. I=∮c-y2dx+xydy The green theorem is defined as, ∮cPdx+Qdy=∬R∂Q∂y−∂P∂x…

Q: Evaluate SfF.ds where F = (3xy², 3x²y, z³) and M M is the surface of the sphere of radius 3 centered…

A: We will use divergence theorem to find surface integral of given vector field  If region R bounded…

Q: Find the highest and lowest point of the surface z2 = x2 + y2 ,on the plane 4x – 3y + 8z = 10 %3D…

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Q: 2. Sketch the cylinder y = x² +1.

A: complete graph provided below with python code Explanation:  code behind above :  import numpy as…

Q: Sketch and identify the surface given by x2 + 4y + 4z2 = 0.

A: Equation of the surface is given by:x2+4y+4z2=0 we need to identity the surface and hence sketch the…

Q: (b). Use Green's Theorem to evaluate SF. dr. F(x, y) = (e³x², 2xy), where C is the triangle from…

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Q: Find F. dr where C is a circle of radius 2 in the plane. x+y+z 9, centered at (4, 3, 2) and oriented…

A: Consider the given equation: And the plane x+y+z=9 centered at 4,3,2.

Q: Evaluate z* + x dS where S is the cylinder x2 + y? = 36 between the planes z = -3 and z = 1,…

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Q: The slope of the surface z = xy² in the x- direction at the point (2, 3) is O 12 O 8 O 1 O 9 O 10

A: The slope of the surface z = f(x,y) in the x-direction at (x0,y0) is given by [∂z/∂x](x0,y0) =…

Q: Closest point is (If there is more than one point, enter a comma-separated list.)

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Q: 3. Find the points on the surface z = = xy + 2 that are closest to the origin.

A: Let g(x,y,z) be the constraint function and f(x,y,z) be the function whose extreme values to…

Q: Evaluate | F-dS, where F(x, y, z) = (3y, –2x, –), %3D and S is the surface that is composed of the…

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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: Consider nonzero real constants α and β. A possible graphic representation for the equation plane αx…

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Q: Let y ==a and z =. Find the intersection of these х x2 surfaces.

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Q: Find the area of the surface x²-6y-2z=0 that lies above the triangle bounded by the lines x= 1, y =…

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Q: Evaluate F. dS, where F = 6xy²i + 6x²yj + 2z°k and S is the surface of the unit sphere.

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Q: Find the Surface area of the portión of the cylinder ya + z² =9 above the rectangle in the xy plane.…

A: Cylinder y2 + z2  = 9,z = 9-y2,0 ≤ x ≤ 2,and-3 ≤ y ≤ 3