15- Evaluate the surface integral f, xy dswhere x is the part of the plaue y = 1-x-z that lies in the first octant

Q: Does there exist a surface x(u, v) such that E=G=1,F=0 and L=1,M=−1,N=0?

A: The objective of the question is to determine if there exists a surface x(u, v) that satisfies the…

Q: Calculate :/ F. ds, where F (x, y, z) = (-6x³ + 2yz)i + (-6y³ + sin¯¹ (3xz)) j+ (2z+cos (3x²y²)) k…

A: Given F(x,y,z)=-6x3+2yzi^+-6y3+sin-13xzj^+2z+cos3x2y2k^ and S is the outward-oriented surface of the…

Q: Express the area A of the cross-section cut from the paraboloid- y by the plane z=k as a function of…

A: Given equation of parabloid x2a2 + y2b2 = zcplane : z = k

Q: Evaluate e fy³dz — 2³dy, where C' is the boundary of the region between the circles æ² + y² = 4 and…

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Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

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Q: Evaluate yºda – x³dy, where C' is the boundary of the region between the circles œ² + 1² = 1 and x²…

A: The given problem is to evaluate the given line integral over the given path C where C is boundary…

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: F. dS, where F(x, y, z) = z²xi + 2) + (x2z + y2)k and S is the top half of the sphere x² + y² + z? =…

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Q: Calculate the line integral fe F. dr, where F = (ry, r? + y? + 22, yz), and C is the boundary of the…

A: Step:-1 Given that a parallelogram with vertices 0, 0, 1, 0, 1, 0, 2, 0, -1, 2, 1, -2 and F→=xy,…

Q: Jse Green's Theorem to evaluate the line integral along the positively oriented closed curve C.…

A: The equation of a straight line joining two points x1, y1 and x2, y2 is given by the formula: y-y1 =…

Q: a) Parameterize the curve of intersection of the surfaces z = y² and x² + y² = 9 so that the…

A: Given surfaces,direction is clockwise when viewed from the above.

Q: Use Green's Theorem to evaluate the line integr y? dx + x • when C is the circle of radius 2…

A: Introduction: Let C be a positively oriented, piecewise smooth, simple, closed curve, and D be the…

Q: Let S be the surface given by the first octant part of the paraboloid z=4-x^2-y^2. Find the area of…

A: We need to determine the surface area of the surface bounded by the paraboloid: z=4-x2-y2 in the…

Q: Determine the area of the surface z = 1 + xy that lies above the unit disk x² + y² ≤ 1.

A: