4. Evaluate the surface integral ff. F.dS by the divergence theorem where F = (3xy2, yx? - y,3zx²Land S is the surface x? + y? < 25, 0

Q: . A donut with major radius R and minor radius r can be represented parametrically by r(u, v) =…

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Q: 4. Let S be the surface in R³ defined by 2² - y² +22² = 9. (b) Does the point (2, 1, 3) lie in the…

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Q: Assume S is the surface x1000+y8000+z1000 = 10¹2 and f = (9+ eyz, x³yz, 3x - 5y - 6 cos(zx)).…

A: Here we use the Divergence theorem for a surface S inclosing a volume V

Q: Find Əz ?х Əz da Əz ду = ab brackets (e.g. x*y rather than xy; 2*(x+y) rather than 2(x+y)). • Put…

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Q: 7. Sketch the surface 15zs9-x' - y² and x'+y' <1

A: To find:

Q: Find an equation for the tangent plane to the surface z = 6x3y2 + 3y at the point P(1, -3). Hint:…

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Q: Q1// Evaluate , F ds where F = yi + xj + 4zk and S is the surface x? + y? = 9 in the first octant…

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Q: Determine the point(s) on the surface z = e(3x² +9y2) at which the tangent plane is parallel to the…

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Q: Find the equation of the tangent plane to the surface y = 2x2 – z2 at the point (0, –1, –1) y+ 4z =…

A: We have to find

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: 4. Find both points on the surface xz – x2 – y? = 8 where the tangent plane is horizontal.

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Q: #6. Use Stokes's Theorem to evaluate , curl F n dS where S is the surface z = 4 - x² - y², z≥ 0 and…

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Q: 1. Consider the surface x² + y² - 2xy = x+3y-z = 0 at the point Po(2, -3, 14). Determine the…

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Q: c) Use Stokes' theorem to evaluate where F(x, y, z) = (x²+y²z) i+(x²+xy²) j+(x²+z²) k, and C is…

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Q: 2. For the surface z = In(y+ Vx² + y²). (b) Find the linear approximation at the point (3,-4).

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Q: 4. Find the area of the surface obtained by rotating the curve x²/3 + y>/3 = 1, 0 <y<1 about y-axis.

A: If the curve defined by the parametric equations x=x(t), y=y(t)  with a≤t≤b , is rotated about the…

Q: Find the absolute maximum and minimum of f(x, y) = 3x + y within the domain x² + y² < 4. 1. Absolute…

A: Let's find.

Q: x² - y²-2²-4x - 2z+3=0

A: The given surface is x2-y2-z2-4x-2z+3=0T is the trace of S using x=2.To find the two cordinate when…

Q: . Use Stokes' Theorem (and only Stokes' Theorem) to evaluate F dr, where F(r,y,z) = (3y, -2x, 3y)…

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Q: Evaluate f(V x F) dS where S is the surface x² + y² +3z² = 1, z ≤0 and F = (y, -x, zx³y²). Let the…

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Q: 1. Let S be the portion of the surface x² + z² = 1 lying in the first octant and bounded by x = 0, y…

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Q: The tangent plane and the normal line of the surface x = y? + z2 + l at (3, 1, -1) have the…

A: Solve the following