Chapter 5

(a) Find all the first partial derivatives of the following functions $f(x,y)$ : (i) $x^{2}y$, (ii) $x^2+y^2+4$, (iii) $\sin (x/y)$, (iv) ${\tan }^{-1}(y/x)$, (v) $r(x,y,z)={(x^2+y^2+z^2)}^{1/2}$. 182(b) For (i), (ii) and (v), find ${\partial }^{2}f/\partial x^2,{\partial }^{2}f/\partial y^2,{\partial }^{2}f/\partial x\partial y$. (c) For (iv) verify that ${\partial }^{2}f/\partial x\partial y={\partial }^{2}f/\partial y\partial x$.

Show that the differential $$df=x^{2}dy-(y^2+xy)dx$$ is not exact, but that $dg={\left(x{y}^2\right)}^{-1}df$ is exact. (a) Show that $$df=y(1+x-x^2)dx+x(x+1)dy$$ is not an exact differential. (b) Find the differential equation that a function $g(x)$ must satisfy if $d\varphi =g(x)df$ is to be an exact differential. Verify that $g(x)=e^{-x}$ is a solution of this equation and deduce the form of $\varphi (x,y).$

(a) Show that $$df=y(1+x-x^2)dx+x(x+1)dy$$ is not an exact differential. (b) Find the differential equation that a function $g(x)$ must satisfy if $d\varphi =g(x)df$ is to be an exact differential. Verify that $g(x)=e^{-x}$ is a solution of this equation and deduce the form of $\varphi (x,y)$.

The function $G(t)$ is defined by $$G(t)=F(x,y)=x^2+y^2+3xy$$ where $x(t)=a{t}^2$ and $y(t)=2at$. Use the chain rule to find the values of $(x,y)$ at which $G(t)$ has stationary values as a function of $t$. Do any of them correspond, to the stationary points of $F(x,y)$ as a function of $x$ and $y$ ?

In the $xy$-plane, new coordinates $s$ and $t$ are defined by $$s=\frac{1}{2}(x+y),t=\frac{1}{2}(x-y)$$ Transform the equation $$\frac{{\partial }^2\varphi }{\partial x^2}-\frac{{\partial }^2\varphi }{\partial y^2}=0$$ into the new coordinates and deduce that its general solution can be written $$\varphi (x,y)=f(x+y)+g(x-y)$$ where $f(u)$ and $g(v)$ are arbitrary functions of $u$ and $v$ respectively. 183

The function $f(x,y)$ satisfies the differential equation $$y\frac{\partial f}{\partial x}+x\frac{\partial f}{\partial y}=0$$ By changing to new variables $u=x^2-y^2$ and $v=2xy$, show that $f$ is, in fact, a function of $x^2-y^2$ only.

If $x=e^{\mu }\cos \theta$ and $y=e^u\sin \theta$, show that $$\frac{{\partial }^2\varphi }{\partial u^2}+\frac{{\hat{\partial }}^2\varphi }{\partial {\theta }^2}=(x^2+y^2)(\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2})$$ where $f(x,y)=\varphi (u,\theta )$.

Find and evaluate the maxima, minima and saddle points of the function $$f(x,y)=xy(x^2+y^2-1)$$

Show that $$f(x,y)=x^3-12xy+48x+b{y}^2,b\ne 0$$ has two, one, or zero stationary points according to whether $|b|$ is less than, equal to, or greater than $3.$

Locate the stationary points of the function $$f(x,y)=(x^2-2y^2)\exp [-(x^2+y^2)/a^2]$$ where $a$ is a non-zero constant. Sketch the function along the $x$ - and $y$-axes and hence identify the nature and values of the stationary points.