91Ó°ÊÓ

Select the correct ansuer or fill up the blanks in each of the following problems: \(\frac{\partial(u, v)}{\partial(x, y)} \times \frac{\partial(x, y)}{\partial(u, v)}\) equaks \(\begin{array}{llll}\text { (a) }-1 & \text { (b) } 1 & \text { (c) zero } & \text { (d) none of these. }\end{array}\)

Select the correct ansuer or fill up the blanks in each of the following problems: \(\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\) is a homogeneous function of degree

Select the correct ansuer or fill up the blanks in each of the following problems: If \(f(x, y)=\frac{1}{x^{3}}+\frac{1}{y^{3}}+\frac{1}{x^{3}+y^{8}}\), then \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{d y}\) is (a) 0 (b) \(3 f\) (c) 9 (d) \(-3 f\).

Select the correct ansuer or fill up the blanks in each of the following problems: If \(u=x^{4}+y^{4}+3 x^{2} y^{2}\), then \(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=\ldots \ldots\)

If \(u\) is a composite function of \(t\), defined by the relations \(u=f(x, y) ; x=\varphi(t), y=y(t)\), then total derivative \(\frac{d u}{d t}\)

If \(u=\cos ^{-1}(x / y)+\tan ^{-1}(y / x)\), then \(x^{2} u_{m}+2 x y u_{2}+y^{2} u_{y y}\) is (a) \(u\) (b) \(2 u\) (c) 0 (d) 1 .

If \(u=f(x+\alpha y)+g(x-a y)\), then \(\frac{\partial^{2} u}{\partial y^{2}}\) equals (a) \(\frac{\partial^{2} u}{\partial x^{2}}\) (b) \(a \frac{\partial^{2} u}{\partial x^{2}}\) (c) \(a^{2} \frac{\partial^{2} u}{d x^{2}}\) (d) \(\frac{\partial^{2} u}{\partial x \partial y}\).