Lila: /* Resources */

2021-11-02T21:28:11Z

Resources

Lila at 15:57, 6 October 2021

2021-10-06T15:57:16Z

Vu.nguyen: Created page with "[https://www.youtube.com/watch?v=CnVes9TdnPo How To Find The Directional Derivative and The Gradient Vector ] Video by The Organic Chemistry Tutor 2019 [https://www.youtube..."

2020-10-26T03:07:19Z

Created page with "*[https://www.youtube.com/watch?v=CnVes9TdnPo How To Find The Directional Derivative and The Gradient Vector ] Video by The Organic Chemistry Tutor 2019 *[https://www.youtube..."

New page

*[https://www.youtube.com/watch?v=CnVes9TdnPo How To Find The Directional Derivative and The Gradient Vector ] Video by The Organic Chemistry Tutor 2019

*[https://www.youtube.com/watch?v=OBELQIPH5xY Implicit Differentiation With Partial Derivatives Using The Implicit Function Theorem] Video by The Organic Chemistry Tutor 2019

← Older revision		Revision as of 21:28, 2 November 2021
Line 69:		Line 69:
	* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Directional_Derivative_and_the_Gradient_Vector Directional Derivative and the Gradient Vector], WikiBooks: Calculus/Multivariable Calculus		* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Directional_Derivative_and_the_Gradient_Vector Directional Derivative and the Gradient Vector], WikiBooks: Calculus/Multivariable Calculus
	* [https://www.youtube.com/watch?v=OBELQIPH5xY Implicit Differentiation With Partial Derivatives Using The Implicit Function Theorem] Video by The Organic Chemistry Tutor 2019		* [https://www.youtube.com/watch?v=OBELQIPH5xY Implicit Differentiation With Partial Derivatives Using The Implicit Function Theorem] Video by The Organic Chemistry Tutor 2019
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Directional_Derivative_and_the_Gradient_Vector Directional Derivative and the Gradient Vector, WikiBooks: Calculus/Multivariable Calculus] under a CC BY-SA license

← Older revision		Revision as of 15:57, 6 October 2021
Line 1:		Line 1:
−	*[https://~~www~~.~~youtube.com/watch?v=CnVes9TdnPo How To Find The Directional Derivative and The Gradient Vector ] Video by The Organic Chemistry Tutor 2019~~	+	=== Directional derivatives===
		+	Normally, a partial derivative of a function with respect to one of its variables, say, ''x''<sub>''j''</sub>, takes the derivative of that "slice" of that function parallel to the ''x''<sub>''j''</sub>'th axis.
		+	<!-- needs pic -->

−	*[https://www.youtube.com/watch?v=OBELQIPH5xY Implicit Differentiation With Partial Derivatives Using The Implicit Function Theorem] Video by The Organic Chemistry Tutor 2019	+	More precisely, we can think of cutting a function '''''f'''''(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) in space along the ''x''<sub>''j''</sub>'th axis, with keeping everything but the ''x''<sub>''j''</sub> variable constant.
		+
		+	From the definition, we have the partial derivative at a point '''''p''''' of the function along this slice as
		+	:<math>{\partial \mathbf{f} \over \partial x_j} = \lim_{t\rightarrow 0} {\mathbf{f}(\mathbf{p}+t\mathbf{e}_j) - \mathbf{f}(\mathbf{p}) \over t}</math>
		+
		+	provided this limit exists.
		+
		+	Instead of the basis vector, which corresponds to taking the derivative along that axis, we can pick a vector in any direction (which we usually take as being a unit vector), and we take the ''directional derivative'' of a function as
		+	:<math>{\partial \mathbf{f} \over \partial \mathbf{d}} = \lim_{t\rightarrow 0} {\mathbf{f}(\mathbf{p}+t\mathbf{d}) - \mathbf{f}(\mathbf{p}) \over t}</math>
		+	where '''''d''''' is the direction vector.
		+
		+	If we want to calculate directional derivatives, calculating them from the limit definition is rather painful, but, we have the following: if '''''f''''' : '''R'''<Sup>''n''</sup> → '''R''' is differentiable at a point '''''p''''', \|'''''p'''''\|=1,
		+	: <math>{\partial \mathbf{f} \over \partial \mathbf{d}} = D_\mathbf{p} \mathbf{f}(\mathbf{d})</math>
		+
		+	There is a closely related formulation which we'll look at in the next section.
		+
		+	=== Gradient vectors ===
		+	The partial derivatives of a scalar tell us how much it changes if we move along one of the axes. What if we move in a different direction?
		+
		+	We'll call the scalar ''f'', and consider what happens if we move an infintesimal direction '''''dr'''''=(''dx,dy,dz''), using the chain rule.
		+	:<math>\mathbf{df}=dx\frac{\partial f}{\partial x} +
		+	dy\frac{\partial f}{\partial y}+dz\frac{\partial f}{\partial z}</math>
		+
		+	This is the dot product of '''dr''' with a vector whose components are the partial derivatives of '''''f''''', called the gradient of '''''f'''''
		+
		+	<math>\operatorname{grad} \mathbf{f} = \nabla \mathbf{f} =
		+	\left(\frac{\partial \mathbf{f}(\mathbf{p})}{\partial x_1},\cdots,
		+	\frac{\partial \mathbf{f}(\mathbf{p})}{\partial x_n}\right)</math>
		+
		+	We can form directional derivatives at a point '''''p''''', in the direction '''''d''''' then by taking the dot product of the gradient with '''''d'''''
		+	:<math>{\partial \mathbf{f}(\mathbf{p}) \over \partial \mathbf{d}} =\mathbf{d} \cdot \nabla \mathbf{f}(\mathbf{p})</math>.
		+
		+	Notice that grad ''f'' looks like a vector multiplied by a scalar. This particular combination of partial derivatives is commonplace, so we abbreviate it to
		+	:<math>\nabla = \left( \frac{\partial }{\partial x},
		+	\frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right) </math>
		+
		+	We can write the action of taking the gradient vector by writing this as an ''operator''. Recall that in the one-variable case we can write ''d''/''dx'' for the action of taking the derivative with respect to ''x''. This case is similar, but '''∇''' acts like a vector.
		+
		+	We can also write the action of taking the gradient vector as:
		+	:<math>\nabla = \left( \frac{\partial }{\partial x_1},
		+	\frac{\partial }{\partial x_2}, \cdots \frac{\partial }{\partial x_n}\right) </math>
		+
		+	==== Properties of the gradient vector ====
		+	===== Geometry =====
		+	* Grad '''''f'''''('''''p''''') is a vector pointing in the direction of steepest slope of '''''f'''''. \|grad '''''f'''''('''''p''''')\| is the rate of change of that slope at that point.
		+
		+	For example, if we consider h(''x'', ''y'')=''x''<sup>2</sup>+''y''<sup>2</sup>. The level sets of ''h'' are concentric circles, centred on the origin, and
		+	:<math> \nabla h = (h_x,h_y) = 2(x,y)= 2 \mathbf{r}</math>
		+	grad ''h'' points directly away from the origin, at right angles to the contours.
		+
		+	* Along a level set, (∇'''''f''''')('''''p''''') is perpendicular to the level set {'''''x'''''\|'''''f'''''('''''x''''')='''''f'''''('''''p''''') at '''''x'''''='''''p'''''}.
		+
		+	If '''dr''' points along the contours of ''f'', where the function is constant, then ''df'' will be zero. Since ''df'' is a dot product, that means that the two vectors, '''df''' and grad ''f'', must be at right angles, i.e. the gradient is at right angles to the contours.
		+
		+	===== Algebraic properties =====
		+	Like ''d/dx'', ∇ is linear. For any pair of constants, ''a'' and ''b'', and any pair of scalar functions, ''f'' and ''g''
		+	:<math>\frac{d}{dx} (af+bg)= a\frac{d}{dx}f + b\frac{d}{dx}g
		+	\quad \nabla (af+bg) = a \nabla f + b \nabla g</math>
		+
		+	Since it's a vector, we can try taking its dot and cross product with other vectors, and with itself.
		+
		+
		+	==Resources==
		+	* [https://www.youtube.com/watch?v=CnVes9TdnPo How To Find The Directional Derivative and The Gradient Vector ] Video by The Organic Chemistry Tutor 2019
		+	* [https://en.wikibooks.org/wiki/Calculus/Multivariable_Calculus/Directional_Derivative_and_the_Gradient_Vector Directional Derivative and the Gradient Vector], WikiBooks: Calculus/Multivariable Calculus
		+	* [https://www.youtube.com/watch?v=OBELQIPH5xY Implicit Differentiation With Partial Derivatives Using The Implicit Function Theorem] Video by The Organic Chemistry Tutor 2019

Directional Derivatives and Gradient Vectors - Revision history

Lila: /* Resources */

Lila at 15:57, 6 October 2021

Vu.nguyen: Created page with "*[https://www.youtube.com/watch?v=CnVes9TdnPo How To Find The Directional Derivative and The Gradient Vector ] Video by The Organic Chemistry Tutor 2019 *[https://www.youtube..."

Vu.nguyen: Created page with "[https://www.youtube.com/watch?v=CnVes9TdnPo How To Find The Directional Derivative and The Gradient Vector ] Video by The Organic Chemistry Tutor 2019 [https://www.youtube..."