Multiple Integrals and Vector Calculus (PS2): Coordinate transformations and integrals

Addition (May 5, 2013): More details on completing the integration in Q6, particularly by projecting onto the xy-plane.

5. Exact differentials

The thermodynamic relation $\delta q = C_V \, \de{T} + (RT/V) \, \de{V}$ is not an exact differential. Show that by dividing this equation by $T$, it becomes exact.

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6. Surface integrals

Show that the surface area of a curved portion of a hemisphere of radius $a$ is $2\pi a^2$. Do this in two ways:

(i) Directly integrating the surface area elements of the hemisphere

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(ii) Projecting onto an integral taken over the xy-plane

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7. Surface and volume integrals

a) Find the area of the plane $x - 2y + 5z = 13$ cut out by the cylinder $x^2 + y^2 = 9$.

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b) A uniform lamina is made of that part of the plane $x + y + z = 1$ that lies in the first octant. Find by integration its area and its centre of mass.

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