Chapter 4: Partial Differentiation

Here are some other ways of obtaining the formula in Example 2.

a) Combine the two fractions to get$\mathbf{(}\mathbf{2}\mathit{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{/}\left[n^2(n+1)\right]$. Then note that for large n,$\mathbf{2}\mathit{n}\mathbf{+}\mathbf{1}\mathbf{\cong }\mathbf{2}\mathit{n}$and$\mathit{n}\mathbf{+}\mathbf{1}\mathbf{\cong }\mathit{n}$.

b) Factor the expression as $\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{2}}}\left(1-\frac{1}{{\left(1+{\displaystyle \frac{1}{n+1}}\right)}^2}\right)$, expand${\mathbf{(}\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{)}}^{\mathbf{-}\mathbf{2}}$ by binomial series to two terms, and then simplify.

If${\mathit{P}}^{\mathbf{3}}\mathbf{+}\mathit{s}\mathit{q}\mathbf{=}\mathit{t}$and${\mathit{q}}^{\mathbf{3}}\mathbf{+}\mathit{t}\mathit{p}\mathbf{=}\mathit{s}$, find${\left(\frac{\partial p}{\partial s}\right)}_{\mathbf{t}}\mathbf{,}{\left(\frac{\partial p}{\partial s}\right)}_{\mathbf{q}}$at$\left(p,q,r,s\right)\mathbf{=}\left(-1,2,3,4\right)$.

If,$\mathit{z}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{2}}\mathbf{,}\mathit{x}\mathbf{=}\mathit{r}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{,}\mathbf{}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathit{r}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }$, find the following partial derivatives.

${\left(\frac{\partial z}{\partial r}\right)}_{\mathbf{x}}$

A function $\mathit{f}\left(x,y,z\right)$is called homogeneous of degree n if$\mathit{f}\left(tx,ty,tz\right)\mathbf{=}{\mathit{t}}^{\mathbf{n}}\mathit{f}\left(x,y,z\right)$ . For example${\mathit{z}}^{\mathbf{2}}\mathbf{\text{ln}}\left(x/y\right)$, is homogeneous of degree 2 since

${\left(tz\right)}^{\mathbf{2}}\mathbf{\text{ln}}\frac{\mathbf{t}\mathbf{x}}{\mathbf{t}\mathbf{y}}\mathbf{=}{\mathit{t}}^{\mathbf{2}}\left(z^2\text{ln}\frac{x}{y}\right)$.

Euler’s theorem on homogeneous functions says that of is homogeneous of degree n , then

.$\mathit{x}\frac{\mathbf{\partial }\mathbf{f}}{\mathbf{\partial }\mathbf{x}}\mathbf{+}\mathit{y}\frac{\mathbf{\partial }\mathbf{f}}{\mathbf{\partial }\mathbf{y}}\mathbf{+}\mathit{z}\frac{\mathbf{\partial }\mathbf{f}}{\mathbf{\partial }\mathbf{z}}\mathbf{=}\mathit{n}\mathit{f}$

Prove this theorem.

Question: Show that satisfies $\mathit{u}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\frac{\mathbf{y}}{\mathbf{\pi }}{\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}\frac{\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{d}\mathbf{t}}{{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{t}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}\mathbf{satisfies}\mathbf{}{\mathit{u}}_{\mathbf{x}\mathbf{x}}\mathbf{+}{\mathit{u}}_{\mathbf{y}\mathbf{y}}\mathbf{=}\mathbf{0}$.

If,$\mathit{z}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{2}}\mathbf{,}\mathit{x}\mathbf{=}\mathit{r}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\mathbf{,}\mathbf{}\mathit{y}\mathbf{=}\mathit{r}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }$, find the following partial derivatives.

${\left(\frac{\partial z}{\partial r}\right)}_{\mathbf{y}}$

Find by the Lagrange multiplier method the largest value of the product of three positive numbers if their sum is 1.

If,$\mathit{z}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{2}}\mathbf{,}\mathit{x}\mathbf{=}\mathit{r}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\mathbf{,}\mathbf{}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathit{r}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }$, find the following partial derivatives.

$\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }\mathbf{r}\mathbf{\partial }\mathbf{y}}$

A function $\mathit{f}\left(x,y,z\right)$ is called homogeneous of degree n if $\mathit{f}\left(tx,ty,tz\right)\mathbf{=}{\mathit{t}}^{\mathbf{n}}\mathit{f}\left(x,y,z\right)$. For example, ${\mathit{z}}^{\mathbf{2}}\mathit{l}\mathit{n}\left(x/y\right)$is homogeneous of a degree 2 since

${\left(tz\right)}^{\mathbf{2}}\mathbf{\text{ln}}\frac{\mathbf{t}\mathbf{x}}{\mathbf{t}\mathbf{y}}\mathbf{=}{\mathit{t}}^{\mathbf{2}}\left(z^2\text{ln}\frac{x}{y}\right)$.

Euler’s theorem on homogeneous functions says that of f is homogeneous of degree, then

$\mathit{x}\frac{\mathbf{\partial }\mathbf{f}}{\mathbf{\partial }\mathbf{x}}\mathbf{+}\mathit{y}\frac{\mathbf{\partial }\mathbf{f}}{\mathbf{\partial }\mathbf{y}}\mathbf{+}\mathit{z}\frac{\mathbf{\partial }\mathbf{f}}{\mathbf{\partial }\mathbf{z}}\mathbf{=}\mathit{n}\mathit{f}$.

Prove this theorem.

What proportions will maximize the area shown in the figure (rectangle with isosceles triangles at its ends) if the perimeter is given?