Essentials of Calculus

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X 2 y>dx + bx*ydy. 4 . T dx + x log xdy 5. (x 2 — y) dx — x dy. X 6. O 2 - 2 xy — y) dx - (x 2 — 2 xy + x) dy.
7. (x 2 — axy + y 2 ) dx + (y 2 + xy — ax 2 ) dy.
8. (sin y — e x y) dx + (x cos y — e x ) dy.
9. Show a geometrical interpretation of the differential y dx — x dy, and from purely geometrical considerations show that the integral ( 2 ' 2 (ydx — xdy) must depend upon the path. ** Vl 10. For a gas that follows the law pv = BT, we have I = p, whence dQ = cdT + pdv. Show that while dQ is
...not an exact differential, — & is an exact differential.
/»2, 2 11. Find the value of the integral \ (ydx — x dy) for the following paths between (0, 0) and (2, 2) : (a) A path made up of the straight line joining (0, 0) to (2, 0) and the straight line joining (2, 0) to (2, 2). (&) A path made up of the F-axis from (0, 0) to (0, 2) and the line joining (0, 2) to (2, 2). (c) The curve y = x 3 - 3 x.
12. Find the value of the integral \ (2 xy dx+x 2 dy). Show by choos- «/o, o ing two or more paths that the integral is independent of the path.


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