Partial Derivatives and Integrals

Partial Derivatives of Functions from Rn to Rm

One of the core concepts of multivariable calculus involves the various differentiations of functions from $\mathbb {R} ^{n}$ to $\mathbb {R} ^{m}$. We begin by defining the concept of a partial derivative of such functions.

Definition: Let $S\subseteq \mathbb {R} ^{n}$ be open, $\mathbf {c} \in S$, and $\mathbf {f} :S\to \mathbb {R} ^{m}$. Denote $\mathbf {e} _{k}=(0,0,...,0,\underbrace {1} _{k^{\mathrm {th} }\;coordinate},0,...,0)\in \mathbb {R} ^{n}$ for each $k\in \{1,2,...,n\}$, i.e., $\mathbf {e} _{k}$ is the unit vector in the direction of the $k^{\mathrm {th} }$ coordinate axis. Then the Partial Derivative of $\mathbf {f}$ at $\mathbf {c}$ with Respect to the $k^{\mathrm {th} }$ Variable is defined as $\displaystyle {D_{k}\mathbf {f} (\mathbf {c} )=\lim _{h\to 0}{\frac {\mathbf {f} (\mathbf {c} +h\mathbf {e} _{k})-\mathbf {f} (\mathbf {c} )}{h}}}$ provided that this limit exists.

Suppose that $S\subseteq \mathbb {R} ^{n}$ is open, $\mathbf {c} \in S$, and $f:S\to \mathbb {R}$. Then the partial derivative of $f$ at $\mathbf {c}$ with respect to the $k^{\mathrm {th} }$ variable is:

${\begin{aligned}\quad D_{k}f(\mathbf {c} )&=\lim _{h\to 0}{\frac {f(\mathbf {c} +h\mathbf {e} _{k})-f(\mathbf {c} )}{h}}\\&=\lim _{h\to 0}{\frac {f(c_{1},c_{2},...,c_{k}+h,...,c_{n})-f(c_{1},c_{2},...,c_{k},...c_{n})}{h}}\end{aligned}}$

For example, consider the function $f:\mathbb {R} ^{3}\to \mathbb {R}$ defined by:

${\begin{aligned}\quad f(x,y,z)=3x^{2}yz\end{aligned}}$

Then the partial derivative of $f$ with respect to the variable $x$ at the point $(1,2,-1)$ is:

${\begin{aligned}\quad D_{1}f(1,2,-1)&=\lim _{h\to 0}{\frac {f(1+h,2,-1)-f(1,2,-1)}{h}}\\&=\lim _{h\to 0}{\frac {3(1+h)^{2}(2)(-1)-3(1)^{2}(2)(-1)}{h}}\\&=\lim _{h\to 0}{\frac {-6(1+h)^{2}+6}{h}}\\&=\lim _{h\to 0}{\frac {-6(1+2h+h^{2})+6}{h}}\\&=\lim _{h\to 0}{\frac {-12h-6h^{2}}{h}}\\&=\lim _{h\to 0}-12-6h\\&=-12\end{aligned}}$

We can also easily calculate the partial derivatives $D_{2}f(1,2,-1)$ and $D_{3}(1,2,-1)$. So the definition of a partial derivative for $\mathbf {f} :S\to \mathbb {R} ^{m}$ is somewhat justified since the case when $m=1$ yields the definition of the partial derivative for a multivariable real-valued function.

Furthermore, suppose that $S\subseteq \mathbb {R}$ and that $\mathbf {f} :S\to \mathbb {R} ^{m}$. Then $\mathbf {f} =(f_{1},f_{2},...,f_{m})$ where $f_{i}:S\to \mathbb {R}$ for each $i\in \{1,2,...,m\}$ are single-variable real-valued functions. The partial derivative of $\mathbf {f}$ with respect to the first variable (the only variable, or simply just the derivative) at $\mathbf {c}$ is:

${\begin{aligned}\quad D_{1}\mathbf {f} (c)&=\lim _{h\to 0}{\frac {\mathbf {f} (c+h)-\mathbf {f} (c)}{h}}\\&=\lim _{h\to 0}{\frac {(f_{1}(c+h),f_{2}(c+h),...,f_{m}(c+h))-(f_{1}(c),f_{2}(c),...,f_{m}(c))}{h}}\\&=\lim _{h\to 0}{\frac {(f_{1}(c+h)-f_{1}(c),f_{2}(c+h)-f_{2}(c),...,f_{m}(c+h)-f(c))}{h}}\\&=\left(\lim _{h\to 0}{\frac {f_{1}(c+h)-f_{1}(c)}{h}},\lim _{h\to 0}{\frac {f_{2}(c+h)-f_{2}(c)}{h}},...,\lim _{h\to 0}{\frac {f_{m}(c+h)-f(c)}{h}}\right)\\&=(f_{1}'(c),f_{2}'(c),...,f_{m}'(c))\end{aligned}}$

For example, consider the function $f:\mathbb {R} \to \mathbb {R} ^{4}$ defined by:

${\begin{aligned}\quad \mathbf {f} (t)=(t,t^{2},t^{3},t^{4})\end{aligned}}$

Then the derivative of $\mathbf {f}$ is:

${\begin{aligned}\quad \mathbf {f} (t)=(1,2t,3t^{2},4t^{3})\end{aligned}}$

And the derivative of $\mathbf {f}$ at $c=2$ is:

${\begin{aligned}\quad \mathbf {f} (2)=(1,4,12,32)\end{aligned}}$

Once again, the definition is justified since when $n=1$ we have that the definition reduces down to the special case of differentiating a single variable vector-valued function.

Now let's look at a more complicated example of computing a partial derivative. Let $\mathbf {f} :\mathbb {R} ^{2}\to \mathbb {R} ^{2}$ be defined by:

${\begin{aligned}\quad \mathbf {f} (x,y)=(x^{2}+y^{2},2xy)\end{aligned}}$

Then the partial derivative of $\mathbf {f}$ at $\mathbf {c}$ with respect to the first variable is:

${\begin{aligned}\quad D_{1}\mathbf {f} (\mathbf {c} )&=\lim _{h\to 0}{\frac {\mathbf {f} (\mathbf {c} +h(1,0))-\mathbf {f} (\mathbf {c} )}{h}}\\&=\lim _{h\to 0}{\frac {(f_{1}((c_{1},c_{2})+h(1,0)),f_{2}((c_{1},c_{2})+h(1,0)))-(f_{1}(\mathbf {c} ),f_{2}(c_{1},c_{2})))}{h}}\\&=\lim _{h\to 0}{\frac {f_{1}(c_{1}+h,c_{2}),f_{2}(c_{1}+h,c_{2}))-(f_{1}(c_{1},c_{2}),f_{2}(c_{1},c_{2}))}{h}}\\&=\lim _{h\to 0}{\frac {(f_{1}(c_{1}+h,c_{2})-f_{1}(c_{1},c_{2}),f_{2}(c_{1}+h,c_{2})-f_{2}(c_{1},c_{2}))}{h}}\\&=\lim _{h\to 0}{\frac {((c_{1}+h)^{2}+c_{2}^{2}-[c_{1}^{2}+c_{2}^{2}],2(c_{1}+h)(c_{2})-2c_{1}c_{2})}{h}}\\&=\lim _{h\to 0}{\frac {(c_{1}^{2}+2c_{1}h+h^{2})+c_{2}^{2}-c_{1}^{2}-c_{2}^{2},2c_{1}c_{2}+2c_{2}h-2c_{1}c_{2})}{h}}\\&=\lim _{h\to 0}{\frac {(2c_{1}h+h^{2},2c_{2}h)}{h}}\\&=\left(\lim _{h\to 0}{\frac {2c_{1}h+h^{2}}{h}},\lim _{h\to 0}{\frac {2c_{2}h}{h}}\right)\\&=\left(\lim _{h\to 0}[2c_{1}+h],\lim _{h\to 0}2c_{2}\right)\\&=(2c_{1},2c_{2})\end{aligned}}$

So the partial derivative of $\mathbf {f}$ with respect to the first variable at say $(1,2)$ is $D_{1}\mathbf {f} (1,2)=(2,4)$.

The Riemann Integral

Let $f$ be a bounded function defined on the closed and bounded interval $[a,b]$. Let $P=\{a=x_{0},x_{1},...,x_{n}=b\}$ be a partition of $[a,b]$ with:

${\begin{aligned}\quad a=x_{0}<x_{1}<...<x_{n}=b\end{aligned}}$

Let ${\mathcal {P}}[a,b]$ denote the set of all partitions on $[a,b]$. For each $i\in \{1,2,...,n\}$ we define:

${\begin{aligned}\quad M_{i}&=\sup\{f(x):x\in [x_{i-1},x_{i}]\}\\\quad m_{i}&=\inf\{f(x):x\in [x_{i-1},x_{i}]\}\end{aligned}}$

With the notation above we can define the upper and lower Riemann sums associated with the partition $P$ for the function $f$.

Definition: Let $f$ be a bounded function on the closed and bounded interval $[a,b]$ and let $P\in {\mathcal {\wp }}[a,b]$. The Upper Riemann Sum Associated with the Partition $P$ for $f$ is $\displaystyle {U(P,f)=\sum _{i=1}^{n}M_{i}\Delta x_{i}}$. The Lower Riemann Sum Associated with the Partition $P$ for $f$ is $\displaystyle {L(P,f)=\sum _{i=1}^{n}m_{i}\Delta x_{i}}$.

It can be shown that for any partitions $P_{1},P_{2}\in {\mathcal {\wp }}[a,b]$ with $P_{1}\subseteq P_{2}$ we have that:

${\begin{aligned}\quad U(P_{2},f)\leq U(P_{1},f)\quad \mathrm {and} \quad L(P_{1},f)\leq L(P_{2},f)\end{aligned}}$

So as we consider partitions that are finer and finer, $U(P,f)$ decreases and $L(P,f)$ increases. It can also be shown that for any partitions $P,P'\in \wp [a,b]$:

${\begin{aligned}\quad L(P,f)\leq U(P',f)\end{aligned}}$

That is, the set of all upper Riemann sums is bounded below by any lower Riemann sum, and the set of all lower Riemann sums is bounded above by any upper Riemann sum. We now define the upper and lower Riemann integrals of a bounded function $f$ on $[a,b]$.

Definition: Let $f$ be a bounded function on the closed and bounded interval $[a,b]$. The Upper Riemann Integral of $f$ is defined to be

$(R){\overline {\int }}_{a}^{b}f(x)\,dx=\inf\{U(P,f):P\in \wp [a,b]\}$ and the Lower Riemann Integral of $f$ is defined to be

$(R){\underline {\int }}_{a}^{b}f(x)\,dx=\sup\{L(P,f):P\in \wp [a,b]\}$.

Another way t o define the upper and lower Riemann integrals of $f$ is through step functions. If $\varphi$ is a step function defined on $[a,b]$ then it is easy to show that the upper and lower Riemann integrals of $\varphi$ exist and define the upper and lower Riemann integrals of $f$ to also be:

${\begin{aligned}\quad (R){\overline {\int }}_{a}^{b}f(x)\,dx=\inf \left\{(R)\int _{a}^{b}\psi (x)\,dx:\psi {\text{is a step function}},f(x)\leq \psi (x){\text{on}}[a,b]\right\}\end{aligned}}$

${\begin{aligned}\quad (R){\underline {\int }}_{a}^{b}f(x)\,dx=\sup \left\{(R)\int _{a}^{b}\varphi (x)\,dx:\varphi {\text{is a step function}},\varphi (x)\leq f(x){\text{on}}[a,b]\right\}\end{aligned}}$

We are finally able to define what it means for a bounded function $f$ defined on a closed and bounded interval $[a,b]$ to be Riemann integrable.

Definition: Let $f$ be a bounded function on the closed and bounded interval $[a,b]$. Then $f$ is said to be Riemann Integrable on $[a,b]$ denoted $f\in R[a,b]$ if $(R){\overline {\int }}_{a}^{b}f(x)\,dx={\underline {\int }}_{a}^{b}f(x)\,dx$.

Licensing

Content obtained and/or adapted from:

• Partial Derivatives of Functions from Rn to Rm, mathonline.wikidot.com under a CC BY-SA license
• The Riemann Integral, mathonline.wikidot.com under a CC BY-SA license