Partial Derivatives and Integrals - Revision history

Khanh at 00:09, 12 November 2021

2021-11-12T00:09:30Z

Lila: /* Partial Derivatives of Functions from Rn to Rm */

2021-11-10T18:25:29Z

Partial Derivatives of Functions from Rn to Rm

Lila at 18:17, 10 November 2021

2021-11-10T18:17:47Z

Lila: Created page with "===Partial Derivatives of Functions from Rn to Rm===
One of the core concepts of multivariable calculus involves the various differentiations of functions from
2021-11-10T18:17:16Z

Created page with "===Partial Derivatives of Functions from Rn to Rm=== One of the core concepts of multivariable calculus involves the various differentiations of functions from One of the core concepts of multivariable calculus involves the various differentiations of functions from <math>\mathbb{R}^n</math> to <math>\mathbb{R}^m</math>. We begin by defining the concept of a partial derivative of such functions.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
<td>Definition: Let <math>S \subseteq \mathbb{R}^n</math> be open, <math>\mathbf{c} \in S</math>, and <math>\mathbf{f} : S \to \mathbb{R}^m</math>. Denote <math>\mathbf{e}_k = (0, 0, ..., 0, \underbrace{1}_{k^{\mathrm{th}} \: coordinate}, 0, ..., 0) \in \mathbb{R}^n</math> for each <math>k \in \{ 1, 2, ..., n \}</math>, i.e., <math>\mathbf{e}_k</math> is the unit vector in the direction of the <math>k^{\mathrm{th}}</math> coordinate axis. Then the Partial Derivative of <math>\mathbf{f}</math> at <math>\mathbf{c}</math> with Respect to the <math>k^{\mathrm{th}}</math> Variable is defined as <math>\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}}</math> provided that this limit exists.</td>
</blockquote>
Suppose that <math>S \subseteq \mathbb{R}^n</math> is open, <math>\mathbf{c} \in S</math>, and <math>f : S \to \mathbb{R}</math>. Then the partial derivative of <math>f</math> at <math>\mathbf{c}</math> with respect to the <math>k^{\mathrm{th}}</math> variable is:
<div style="text-align: center;"><math>\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}</math></div>
For example, consider the function <math>f : \mathbb{R}^3 \to \mathbb{R}</math> defined by:
<div style="text-align: center;"><math>\begin{align} \quad f(x, y, z) = 3x^2yz \end{align}</math></div>
Then the partial derivative of <math>f</math> with respect to the variable <math>x</math> at the point <math>(1, 2, -1)</math> is:
<div style="text-align: center;"><math>\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}</math></div>
We can also easily calculate the partial derivatives <math>D_2 f(1, 2, -1)</math> and <math>D_3(1, 2, -1)</math>. So the definition of a partial derivative for <math>\mathbf{f} : S \to \mathbb{R}^m</math> is somewhat justified since the case when <math>m = 1</math> yields the definition of the partial derivative for a multivariable real-valued function.
Furthermore, suppose that <math>S \subseteq \mathbb{R}</math> and that <math>\mathbf{f} : S \to \mathbb{R}^m</math>. Then <math>\mathbf{f} = (f_1, f_2, ..., f_m)</math> where <math>f_i : S \to \mathbb{R}</math> for each <math>i \in \{ 1, 2, ..., m \}</math> are single-variable real-valued functions. The partial derivative of <math>\mathbf{f}</math> with respect to the first variable (the only variable, or simply just the derivative) at <math>\mathbf{c}</math> is:
<div style="text-align: center;"><math>\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}</math></div>
For example, consider the function <math>f : \mathbb{R} \to \mathbb{R}^4</math> defined by:
<div style="text-align: center;"><math>\begin{align} \quad \mathbf{f}(t) = (t, t^2, t^3, t^4) \end{align}</math></div>
Then the derivative of <math>\mathbf{f}</math> is:
<div style="text-align: center;"><math>\begin{align} \quad \mathbf{f}(t) = (1, 2t, 3t^2, 4t^3) \end{align}</math></div>
And the derivative of <math>\mathbf{f}</math> at <math>c = 2</math> is:
<div style="text-align: center;"><math>\begin{align} \quad \mathbf{f}(2) = (1, 4, 12, 32) \end{align}</math></div>
Once again, the definition is justified since when <math>n = 1</math> 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 <math>\mathbf{f} : \mathbb{R}^2 \to \mathbb{R}^2</math> be defined by:
<div style="text-align: center;"><math>\begin{align} \quad \mathbf{f}(x, y) = (x^2 + y^2, 2xy) \end{align}</math></div>
Then the partial derivative of <math>\mathbf{f}</math> at <math>\mathbf{c}</math> with respect to the first variable is:
<div style="text-align: center;"><math>\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}</math></div>
So the partial derivative of <math>\mathbf{f}</math> with respect to the first variable at say <math>(1, 2)</math> is <math>D_1 \mathbf{f}(1, 2) = (2, 4)</math>.

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-===Partial Derivatives of Functions from Rn to Rm===
+==Partial Derivatives of Functions from Rn to Rm==
 <p>One of the core concepts of multivariable calculus involves the various differentiations of functions from <span class="math-inline"><math>\mathbb{R}^n</math></span> to <span class="math-inline"><math>\mathbb{R}^m</math></span>. We begin by defining the concept of a partial derivative of such functions.</p>
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
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 <div style="text-align: center;"><math>\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}</math></div>
 <p>So the partial derivative of <span class="math-inline"><math>\mathbf{f}</math></span> with respect to the first variable at say <span class="math-inline"><math>(1, 2)</math></span> is <span class="math-inline"><math>D_1 \mathbf{f}(1, 2) = (2, 4)</math></span>.</p>
 ==Licensing==
 Content obtained and/or adapted from:
 * [http://mathonline.wikidot.com/partial-derivatives-of-functions-from-rn-to-rm Partial Derivatives of Functions from Rn to Rm, mathonline.wikidot.com] under a CC BY-SA license

← Older revision		Revision as of 18:25, 10 November 2021
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	<div style="text-align: center;"><math>\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}</math></div>		<div style="text-align: center;"><math>\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}</math></div>
	<p>So the partial derivative of <span class="math-inline"><math>\mathbf{f}</math></span> with respect to the first variable at say <span class="math-inline"><math>(1, 2)</math></span> is <span class="math-inline"><math>D_1 \mathbf{f}(1, 2) = (2, 4)</math></span>.</p>		<p>So the partial derivative of <span class="math-inline"><math>\mathbf{f}</math></span> with respect to the first variable at say <span class="math-inline"><math>(1, 2)</math></span> is <span class="math-inline"><math>D_1 \mathbf{f}(1, 2) = (2, 4)</math></span>.</p>
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [http://mathonline.wikidot.com/partial-derivatives-of-functions-from-rn-to-rm Partial Derivatives of Functions from Rn to Rm, mathonline.wikidot.com] under a CC BY-SA license

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 <p>One of the core concepts of multivariable calculus involves the various differentiations of functions from <span class="math-inline"><math>\mathbb{R}^n</math></span> to <span class="math-inline"><math>\mathbb{R}^m</math></span>. We begin by defining the concept of a partial derivative of such functions.</p>
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
-<td><strong>Definition:</strong> Let <span class="math-inline"><math>S \subseteq \mathbb{R}^n</math></span> be open, <span class="math-inline"><math>\mathbf{c} \in S</math></span>, and <span class="math-inline"><math>\mathbf{f} : S \to \mathbb{R}^m</math></span>. Denote <span class="math-inline"><math>\mathbf{e}_k = (0, 0, ..., 0, \underbrace{1}_{k^{\mathrm{th}} \: coordinate}, 0, ..., 0) \in \mathbb{R}^n</math></span> for each <span class="math-inline"><math>k \in \{ 1, 2, ..., n \}</math></span>, i.e., <span class="math-inline"><math>\mathbf{e}_k</math></span> is the unit vector in the direction of the <span class="math-inline"><math>k^{\mathrm{th}}</math></span> coordinate axis. Then the <strong>Partial Derivative of <span class="math-inline"><math>\mathbf{f}</math></span> at <span class="math-inline"><math>\mathbf{c}</math></span> with Respect to the <span class="math-inline"><math>k^{\mathrm{th}}</math></span> Variable</strong> is defined as <span class="math-inline"><math>\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}}</math></span> provided that this limit exists.</td>
+<td><strong>Definition:</strong> Let <span class="math-inline"><math>S \subseteq \mathbb{R}^n</math></span> be open, <span class="math-inline"><math>\mathbf{c} \in S</math></span>, and <span class="math-inline"><math>\mathbf{f} : S \to \mathbb{R}^m</math></span>. Denote <span class="math-inline"><math>\mathbf{e}_k = (0, 0, ..., 0, \underbrace{1}_{k^{\mathrm{th}} \; coordinate}, 0, ..., 0) \in \mathbb{R}^n</math></span> for each <span class="math-inline"><math>k \in \{ 1, 2, ..., n \}</math></span>, i.e., <span class="math-inline"><math>\mathbf{e}_k</math></span> is the unit vector in the direction of the <span class="math-inline"><math>k^{\mathrm{th}}</math></span> coordinate axis. Then the <strong>Partial Derivative of <span class="math-inline"><math>\mathbf{f}</math></span> at <span class="math-inline"><math>\mathbf{c}</math></span> with Respect to the <span class="math-inline"><math>k^{\mathrm{th}}</math></span> Variable</strong> is defined as <span class="math-inline"><math>\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}}</math></span> provided that this limit exists.</td>
 </blockquote>
 <p>Suppose that <span class="math-inline"><math>S \subseteq \mathbb{R}^n</math></span> is open, <span class="math-inline"><math>\mathbf{c} \in S</math></span>, and <span class="math-inline"><math>f : S \to \mathbb{R}</math></span>. Then the partial derivative of <span class="math-inline"><math>f</math></span> at <span class="math-inline"><math>\mathbf{c}</math></span> with respect to the <span class="math-inline"><math>k^{\mathrm{th}}</math></span> variable is:</p>