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 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 k^{\mathrm{th}}} coordinate axis. 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 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 k^{\mathrm{th}}} Variable is defined as 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{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 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 S \subseteq \mathbb{R}^n} is open, 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} \in S} , 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 f : S \to \mathbb{R}} . 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 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 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 k^{\mathrm{th}}} variable 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 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{align}}

For example, consider 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 : \mathbb{R}^3 \to \mathbb{R}} 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 f(x, y, z) = 3x^2yz \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 f} with respect to the variable 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 x} at the point 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, -1)} 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 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{align}}

We can also easily calculate the partial derivatives 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_2 f(1, 2, -1)} 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 D_3(1, 2, -1)} . So the definition of a partial derivative 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 \mathbf{f} : S \to \mathbb{R}^m} is somewhat justified since the case 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 m = 1} yields the definition of the partial derivative for a multivariable real-valued function.

Furthermore, suppose 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 S \subseteq \mathbb{R}} and 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 \mathbf{f} : S \to \mathbb{R}^m} . 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 \mathbf{f} = (f_1, f_2, ..., f_m)} where 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_i : S \to \mathbb{R}} 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, ..., m \}} are single-variable real-valued functions. 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}} with respect to the first variable (the only variable, or simply just the derivative) 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}} 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 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{align}}

For example, consider 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 : \mathbb{R} \to \mathbb{R}^4} 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}(t) = (t, t^2, t^3, t^4) \end{align}}

Then 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}} 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:

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 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_1c_2)}{h} \\ &= \lim_{h \to 0} \frac{(c_1^2 + 2c_1h + h^2) + c_2^2 - c_1^2 - c_2^2, 2c_1c_2 + 2c_2h - 2c_1c_2)}{h} \\ &= \lim_{h \to 0} \frac{(2c_1h + h^2, 2c_2h)}{h} \\ &= \left ( \lim_{h \to 0} \frac{2c_1h + h^2}{h}, \lim_{h \to 0} \frac{2c_2h}{h} \right ) \\ &= \left ( \lim_{h \to 0} [2c_1 + h], \lim_{h \to 0} 2c_2 \right ) \\ &= (2c_1, 2c_2) \end{align}}

So 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}} 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 $D_{1}\mathbf {f} (1,2)=(2,4)$.