91Ó°ÊÓ

Given \(f=\sin (u+v)\), where \(v=\cos u\), (i) find \(\frac{d f}{d u}\), (ii) if \(u=e^{-t}\), find \(\frac{d f}{d t}\).

If \(U=U(V, T)\) and \(p=p(V, T)\) are functions of \(V\) and \(T\) and if \(H=U+p V\), show that \(\left(\frac{\partial H}{\partial T}\right)_{p}-\left(\frac{\partial U}{\partial T}\right)_{V}=\left[\left(\frac{\partial U}{\partial V}\right)_{T}+p\right]\left(\frac{\partial V}{\partial T}\right)_{p}\)

Given \(x=a u+b v\) and \(y=b u-a v\), where \(a\) and \(b\) are constants, (i) if \(f\) is a function of \(x\) and \(y\), express \(\left(\frac{\partial f}{\partial u}\right)_{v}\) and \(\left(\frac{\partial f}{\partial v}\right)_{u}\) in terms of \(\left(\frac{\partial f}{\partial x}\right)_{y}\) and \(\left(\frac{\partial f}{\partial y}\right)_{x}\), (ii) if \(f=x^{2}+y^{2}\), find \(\left(\frac{\partial f}{\partial u}\right)_{v}\) and \(\left(\frac{\partial f}{\partial v}\right)_{u}\) in terms of \(u\) and \(v\).

Given \(u=x^{n}+y^{n}\) and \(v=x^{n}-y^{n}\), where \(n\) is a constant, (i) show that \(\left(\frac{\partial x}{\partial u}\right)_{v}\left(\frac{\partial u}{\partial x}\right)_{y}=\frac{1}{2}=\left(\frac{\partial y}{\partial v}\right)_{u}\left(\frac{\partial v}{\partial y}\right)_{x}\). (ii) If \(f\) is a function of \(x\) and \(y\), express \(\left(\frac{\partial f}{\partial x}\right)_{y}\) and \(\left(\frac{\partial f}{\partial y}\right)_{x}\) in terms of \(\left(\frac{\partial f}{\partial u}\right)_{v}\) and \(\left(\frac{\partial f}{\partial v}\right)_{u}\). Hence, (iii) if \(f=u^{2}-v^{2}\), find \(\left(\frac{\partial f}{\partial x}\right)_{y}\) and \(\left(\frac{\partial f}{\partial y}\right)_{x}\) in terms of \(x\) and \(y .\)

Show that the following functions of position in a plane satisfy Laplace's equation: $$ x^{5}-10 x^{3} y^{2}+5 x y^{4} $$

Show that the following functions of position in a plane satisfy Laplace's equation: $$ \left(A r+\frac{B}{r}\right) \sin \theta $$

Show that the following functions of position in a plane satisfy Laplace's equation: $$ r^{n} \cos n \theta, n=1,2,3, \ldots $$

Test for exactness: $$ (4 x+3 y) d x+(3 x+8 y) d y $$

Test for exactness: $$ (6 x+5 y+7) d x+(4 x+10 y+8) d y $$

Test for exactness: $$ y \cos x d x+\sin x d y $$