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Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

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Sec. 10]_Change of Variables_2H<br />

1962. The functions y and z of the <strong>in</strong>dependent variable x are<br />

def<strong>in</strong>ed by the follow<strong>in</strong>g system of equations:<br />

F<strong>in</strong>d dy, dz, d*y, d*z.<br />

1963. The functions u and v of the <strong>in</strong>dependent variables x and y<br />

are def<strong>in</strong>ed implicitly by the system of equations<br />

Calculate<br />

du du d*u d 2 u d z u dv dv 6 2 v d*v d z v<br />

2 '<br />

~dy<br />

'<br />

dx' d~y ~d\*' dxdy j<br />

for x=--Q, y 1.<br />

1964. The functions u and v of the <strong>in</strong>dependent variables x<br />

and y are def<strong>in</strong>ed implicitly by the system of equations<br />

F<strong>in</strong>d du, dv, d*u, d*v.<br />

1965. The functions u and v of the variables x and y are<br />

def<strong>in</strong>ed implicitly by the system of equations<br />

du du du (b<br />

.v = (p(w, v), //=-i|)(w, v).<br />

51' cty' d~x> fy'<br />

1966. a) F<strong>in</strong>d ^ and g? , if x = M cos u, y w s<strong>in</strong> u, 2 =<br />

b) F<strong>in</strong>d and ~ y<br />

c) F<strong>in</strong>d dz, if jc = ,<br />

\l x = u + v, y = u v,z = uv.<br />

,<br />

1967. e = F(r, cp) where r and

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