This question concerns the vector field F = -xy²i-r²yj + z(x² + y²) k. (a) Calculate the divergence of F. Use the divergence theorem to show that the flux of F over any closed surface is equal to zero. (b) A cylinder of height h and radius R has its base in the xy-plane and its axis of symmetry along the z-axis, as shown in the diagram below. X O R h SE Calculate the flux of F over the curved portion S of the surface of the cylinder by using the result of part (a) to relate the flux of F over S to the flux of F over the flat top surface T and the flux of F over the flat bottom surface B. (Hint: Calculating the flux of F over T and B is easier in cylindrical coordinates.) yus od bless (s

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A: We find the required integrtion using fundamental theorem of line Integral.

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Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

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A: Given:

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A: We know that,      1.              2.

Q: Use (a) parametrization; field (b) Stokes' Theorem to compute fF. dr for the vector F = (x²+z)i +…

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A: The surface H  is defined by    .The vector field  is given by   .We have to sketch the surface H.

Q: (e) Verify Green's theorem for a vector field F(x, y) = y²i + x²j and a triangle bounded by the…

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Q: How much work is required to move an object in the vector field F= 5 y2i + 2xy J  Along the curve ,…

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Q: Let C₁ be the directed line segment that starts at (2,0) and ends at (-3,-5) and C₂ be the parabola…

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Q: a) Calculate the flux of the vector field F(x, y, z) = 275k through a sphere of radlus 4 centered at…

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Q: Consider the vector field F(x,y,z) = (3x²y)i + (x³ + 1) + (92²) and the curve C made up of the…

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Q: Find the field lines and sketch the field portrait for the following vector fields F : R? → R². (a)…

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Q: magnitude 1.

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Q: 2x 2y Consider the vector field F = |x² + y2’ x² + y² (a) If C1 is the unit circle x² + y² = 1,…

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Q: 1) Given the vector fields F1 = x²2 and F2 = xâ+ yŷ + z2, calculate the divergence and curl of F1…

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Q: Sketch the given vector fields by drawing some representative vectors. (a) F(x, y) = (x − y) î+xî…

A: Given:  a F→x, y=x-yi^ +xj^b F→x, y=y, -xx2+y2c F→x, y=2yj To sketch the given vector field.