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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mathbf{f}(t) = (1, 2t, 3t^2, 4t^3) \end{align}}

And the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{f}} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = 2} is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mathbf{f}(2) = (1, 4, 12, 32) \end{align}}

Once again, the definition is justified since when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{f} : \mathbb{R}^2 \to \mathbb{R}^2} be defined by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mathbf{f}(x, y) = (x^2 + y^2, 2xy) \end{align}}

Then the partial derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{f}} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1, 2)} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1 \mathbf{f}(1, 2) = (2, 4)} .

The Riemann Integral

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} be a bounded function defined on the closed and bounded interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \{ a = x_0, x_1, ..., x_n = b \}} be a partition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} with:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a = x_0 < x_1 < ... < x_n = b \end{align}}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal P[a, b]} denote the set of all partitions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} . For each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in \{ 1, 2, ..., n \}} we define:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

With the notation above we can define the upper and lower Riemann sums associated with the partition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} for the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} .

Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} be a bounded function on the closed and bounded interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \in \mathcal \wp [a, b]} . The Upper Riemann Sum Associated with the Partition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{U(P, f) = \sum_{i=1}^{n} M_i \Delta x_i}} . The Lower Riemann Sum Associated with the Partition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{L(P, f) = \sum_{i=1}^{n} m_i \Delta x_i}} .

It can be shown that for any partitions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1, P_2 \in \mathcal \wp [a, b]} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1 \subseteq P_2} we have that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad U(P_2, f) \leq U(P_1, f) \quad \mathrm{and} \quad L(P_1, f) \leq L(P_2, f) \end{align}}

So as we consider partitions that are finer and finer, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(P, f)} decreases and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(P, f)} increases. It can also be shown that for any partitions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P, P' \in \wp [a, b]} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad L(P, f) \leq U(P', f) \end{align}}

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} .

Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} be a bounded function on the closed and bounded interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} . The Upper Riemann Integral of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is defined to be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (R) \overline \int_{a}^{b} f(x) \, dx = \inf \{ U(P, f) : P \in \wp [a, b] \} } and the Lower Riemann Integral of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is defined to be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is through step functions. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} is a step function defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} then it is easy to show that the upper and lower Riemann integrals of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} exist and define the upper and lower Riemann integrals of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} to also be:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

We are finally able to define what it means for a bounded function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} defined on a closed and bounded interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} to be Riemann integrable.

Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} be a bounded function on the closed and bounded interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is said to be Riemann Integrable on $[a,b]$ denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \in R[a, b]} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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