Differentiability - Revision history

Lila: /* Resources */

2021-11-02T21:35:36Z

Resources

Lila at 17:37, 6 October 2021

2021-10-06T17:37:19Z

Lila: /* Jacobian matrix and partial derivatives */

2021-10-06T17:35:52Z

Jacobian matrix and partial derivatives

Lila at 17:34, 6 October 2021

2021-10-06T17:34:39Z

Khanh: Created page with "*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~..."

2021-09-26T04:50:25Z

Created page with "*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~..."

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*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~:text=1%3A%20The%20tangent%20plane%20to,be%20differentiable%20at%20that%20point. Tangent Planes and Differentials], Mathematics LibreTexts

← Older revision		Revision as of 21:35, 2 November 2021
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	* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Chain_Rule Chain Rule], WikiBooks: Calculus/Multivariable Calculus		* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Chain_Rule Chain Rule], WikiBooks: Calculus/Multivariable Calculus
	*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~:text=1%3A%20The%20tangent%20plane%20to,be%20differentiable%20at%20that%20point. Tangent Planes and Differentials], Mathematics LibreTexts		*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~:text=1%3A%20The%20tangent%20plane%20to,be%20differentiable%20at%20that%20point. Tangent Planes and Differentials], Mathematics LibreTexts
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Partial_Derivatives Partial Derivatives, WikiBooks: Calculus/Multivariable Calculus] under a CC BY-SA license
		+	* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Chain_Rule Chain Rule, WikiBooks: Calculus/Multivariable Calculus] under a CC BY-SA license

← Older revision		Revision as of 17:37, 6 October 2021
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	==Resources==		==Resources==
		+	* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Partial_Derivatives Partial Derivatives], WikiBooks: Calculus/Multivariable Calculus
		+	* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Chain_Rule Chain Rule], WikiBooks: Calculus/Multivariable Calculus
	*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~:text=1%3A%20The%20tangent%20plane%20to,be%20differentiable%20at%20that%20point. Tangent Planes and Differentials], Mathematics LibreTexts		*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~:text=1%3A%20The%20tangent%20plane%20to,be%20differentiable%20at%20that%20point. Tangent Planes and Differentials], Mathematics LibreTexts

← Older revision		Revision as of 17:35, 6 October 2021
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	However, it is possible that all the partial derivatives of a function exist at some point yet that function is not differentiable there, so it is very important not to mix derivative (linear map) with the Jacobian (matrix) especially in situations akin to the one cited.		However, it is possible that all the partial derivatives of a function exist at some point yet that function is not differentiable there, so it is very important not to mix derivative (linear map) with the Jacobian (matrix) especially in situations akin to the one cited.
		+
		+	'''Rules of taking Jacobians'''
		+	If '''''f''''' : '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>, and ''h''(''x'') : '''R'''<sup>''m''</sup> → '''R''' are differentiable at ''''p'''':
		+	* <math>J_\mathbf{p} (\mathbf{f}+\mathbf{g}) = J_\mathbf{p} \mathbf{f} + J_\mathbf{p} \mathbf{g}</math>
		+	* <math>J_\mathbf{p} (h\mathbf{f}) = hJ_\mathbf{p} \mathbf{f} + \mathbf{f}(\mathbf{p}) J_\mathbf{p} h</math>
		+	* <math>J_\mathbf{p} (\mathbf{f}\cdot \mathbf{g}) = \mathbf{g}^T J_\mathbf{p} \mathbf{f} + \mathbf{f}^T J_\mathbf{p}\mathbf{g}</math>
		+	Important: make sure the order is right - matrix multiplication is not commutative!
		+
		+	==== Chain rule ====
		+	The chain rule for functions of several variables is as follows. For '''''f''''' : '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup> and '''''g''''' : '''R'''<sup>''n''</sup> → '''R'''<sup>''p''</sup>, and '''''g''''' o '''''f''''' differentiable at '''''p''''', then the Jacobian is given by
		+	: <math>\left( J_{\mathbf{f}(\mathbf{p})} \mathbf{g}\right) \left( J_\mathbf{p} \mathbf{f}\right)</math>
		+	Again, we have matrix multiplication, so one must preserve this exact order.
		+	Compositions in one order may be defined, but not necessarily in the other way.

	==== Continuity and differentiability ====		==== Continuity and differentiability ====

← Older revision		Revision as of 17:34, 6 October 2021
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		+	== Differentiable functions ==
		+	We will start from the one-variable definition of the derivative at a point ''p'', namely
		+	: <math>\lim_{x\rightarrow p} {f(x)-f(p) \over x-p} = f'(p)</math>
		+
		+	Let's change above to equivalent form of
		+	: <math>\lim_{x\rightarrow p} {f(x)-f(p)-f'(p)(x-p) \over x-p} = 0</math>
		+	which achieved after pulling f'(p) inside and putting it over a common denominator.
		+
		+	We can't divide by vectors, so this definition can't be immediately extended to the multiple variable case. Nonetheless, we don't have to: the thing we took interest in was the quotient of two small distances (magnitudes), not their other properties (like sign). It's worth noting that 'other' property of vector neglected is its direction. Now we can divide by the absolute value of a vector, so lets rewrite this definition in terms of absolute values
		+	: <math>\lim_{x\rightarrow p} \frac{\left\|f(x)-f(p)-f'(p)(x-p)\right\|}{\left\| x-p\right\|} = 0</math>
		+
		+	Another form of formula above is obtained by letting <math>h=x-p</math> we have <math>x=p+h</math> and if <math>x \rightarrow p</math>, the <math>h = x - p \rightarrow 0</math>, so
		+	: <math>\lim_{h\rightarrow 0} \frac{\left\|f(p+h)-f(p)-f'(p)h\right\|}{\left\| h\right\|} = 0</math>,
		+	where <math>h</math> can be thought of as a 'small change'.
		+
		+	So, how can we use this for the several-variable case?
		+
		+	If we switch all the variables over to vectors and replace the constant (which performs a linear map in one dimension) with a matrix (which denotes also a linear map), we have
		+	: <math>\lim_{\mathbf{x}\rightarrow\mathbf{p}} {\|\mathbf{f}(\mathbf{x})-\mathbf{f}(\mathbf{p})-\mathbf{A}(\mathbf{x}-\mathbf{p})\| \over \|\mathbf{x}-\mathbf{p}\|} = 0</math>
		+	or
		+	: <math>\lim_{\mathbf{h}\rightarrow\mathbf{0}} {\|\mathbf{f}(\mathbf{p}+\mathbf{h})-\mathbf{f}(\mathbf{p})-\mathbf{A}\mathbf{h}\| \over \|\mathbf{h}\|} = 0</math>
		+	If this limit exists for some '''''f''''' : '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>, and there is a linear map '''''A''''' : '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup> (denoted by matrix ''A'' which is ''m''×''n''), we refer to this map as being the derivative and we write it as D<sub>'''''p'''''</sub> '''''f'''''.
		+
		+	A point on terminology - in referring to the action of taking the derivative (giving the linear map '''''A'''''), we write D<sub>'''''p'''''</sub> '''''f''''', but in referring to the matrix ''A'' itself, it is known as the ''Jacobian'' matrix and is also written ''J''<sub>'''''p'''''</sub> '''''f'''''. More on the Jacobian later.
		+
		+	=== Properties ===
		+	There are a number of important properties of this formulation of the derivative.
		+
		+	==== Affine approximations ====
		+	If '''''f''''' is differentiable at '''''p''''' for '''''x''''' close to '''''p''''', \|'''''f'''''('''''x''''')-('''''f'''''('''''p''''')+''A''('''''x'''''-'''''p'''''))\| is small compared to \|'''''x'''''-'''''p'''''\|, which means that '''''f'''''('''''x''''') is approximately equal to '''''f'''''('''''p''''')+''A''('''''x'''''-'''''p''''').
		+
		+	We call an expression of the form ''g''(''x'')+''c'' affine, when ''g''(''x'') is linear and ''c'' is a constant. '''''f'''''('''''p''''')+''A''('''''x'''''-'''''p''''') is an affine approximation to '''''f'''''('''''x''''').
		+
		+	==== Jacobian matrix and partial derivatives====
		+	The Jacobian matrix of a function is in the form
		+	: <math>\left(J_\mathbf{p} \mathbf{f}\right)_{ij} = \left.{\partial f_i \over \partial x_j}\right\|_\mathbf{p}</math>
		+	for a '''''f''''' : '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>, ''J''<sub>'''''p'''''</sub> '''''f'''' is a ''n''×''m'' matrix.
		+
		+	The consequence of this is that if '''''f''''' is differentiable at '''''p''''', ''all'' the partial derivatives of '''''f''''' exist at '''''p'''''.
		+
		+	However, it is possible that all the partial derivatives of a function exist at some point yet that function is not differentiable there, so it is very important not to mix derivative (linear map) with the Jacobian (matrix) especially in situations akin to the one cited.
		+
		+	==== Continuity and differentiability ====
		+	Furthermore, if all the partial derivatives exist, and are continuous in some neighbourhood of a point '''''p''''', then '''''f''''' is differentiable at '''''p'''''. This has the consequence that for a function '''''f''''' which has its component functions built from continuous functions (such as rational functions, differentiable functions or otherwise), '''''f''''' is differentiable everywhere '''''f''''' is defined.
		+
		+	We use the terminology ''continuously differentiable'' for a function differentiable at '''''p''''' which has all its partial derivatives existing and are continuous in some neighbourhood at '''''p'''''.
		+
		+
		+	==Resources==
	*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~:text=1%3A%20The%20tangent%20plane%20to,be%20differentiable%20at%20that%20point. Tangent Planes and Differentials], Mathematics LibreTexts		*[https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials#:~:text=1%3A%20The%20tangent%20plane%20to,be%20differentiable%20at%20that%20point. Tangent Planes and Differentials], Mathematics LibreTexts