Diffusion Mathematics - Revision history

Bradley Monk at 05:22, 10 August 2014

2014-08-10T05:22:42Z

← Older revision		Revision as of 22:22, 9 August 2014
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	[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]][[Category:Math]]		[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]][[Category:Math]][[Category:Neurobiology]]

Bradley Monk at 00:59, 10 August 2014

2014-08-10T00:59:24Z

← Older revision		Revision as of 17:59, 9 August 2014
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	[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]]		[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]][[Category:Math]]

Bradley Monk at 00:33, 4 November 2013

2013-11-04T00:33:31Z

← Older revision		Revision as of 17:33, 3 November 2013
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	In no particular order, here are some common mathematical concepts often found in diffusion modeling. To simulate and quantify simple diffusion (e.g. 2D Brownian Motion) only requires a working knowledge of basic statistics and trigonometry. Modeling complex forms of diffusion requires the use of vector calculus and differential geometry. There is a thin boundary between simple and complex diffusion and the math becomes dense very quickly. For example, using just a few lines of code, one can easily simulate stochastic diffusion on a flat 2D surface:		In no particular order, here are some common mathematical concepts often found in diffusion modeling. To simulate and quantify simple diffusion (e.g. 2D Brownian Motion) only requires a working knowledge of basic statistics and trigonometry. Modeling complex forms of diffusion requires the use of integral and vector calculus and differential geometry. There is a thin boundary between simple and complex diffusion and the math becomes dense very quickly. For example, using just a few lines of code, one can easily simulate stochastic diffusion on a flat 2D surface:



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Bradley Monk: /* Partial Derivatives */

2013-11-03T23:06:33Z

Partial Derivatives

← Older revision		Revision as of 16:06, 3 November 2013
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	[[File:Partial derivative eq1.png]]		[[File:Partial derivative eq1.png]]

			Read as: del F with respect to X equals...

	Suppose that ƒ is a function of more than one variable. For instance:		Suppose that ƒ is a function of more than one variable. For instance:

Bradley Monk: /* Laplace Operator */

2013-11-03T23:04:27Z

Laplace Operator

← Older revision		Revision as of 16:04, 3 November 2013
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	In mathematics the [http://en.wikipedia.org/wiki/Laplacian Laplace operator] or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows.		In mathematics the [http://en.wikipedia.org/wiki/Laplacian Laplace operator] or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the nabla symbols ∇·∇, ∇2 or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows.

	{{ExpandBox\|more info or media\|		{{ExpandBox\|more info or media\|
	No additional media		No additional media
	}}		}}



	==Gradients==		==Gradients==

Bradley Monk: /* Partial Derivatives */

2013-11-03T22:53:31Z

Partial Derivatives

← Older revision		Revision as of 15:53, 3 November 2013
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	To find the slope of the line tangent to the function at P(1, 1, 3) that is parallel to the xz-plane, the y variable is treated as constant. The graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane y = 1. By finding the derivative of the equation while assuming that y is a constant, the slope of ƒ at the point (x, y, z) is found to be:		To find the slope of the line tangent to the function at P(1, 1, 3) that is parallel to the xz-plane, the y variable is treated as constant. The graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane y = 1. By finding the derivative of the equation while assuming that y is a constant, the slope of ƒ at the point (x, y, z) is found to be:

	[[File:Partial derivative ~~eq3~~.png]]		[[File:Partial derivative eq4.png]]

	So at (1, 1, 3), by substitution, the slope is 3. Therefore		So at (1, 1, 3), by substitution, the slope is 3. Therefore

	[[File:Partial derivative ~~eq4~~.png]]		[[File:Partial derivative eq3.png]]

	at the point (1, 1, 3). That is, the partial derivative of z with respect to x at (1, 1, 3) is 3.		at the point (1, 1, 3). That is, the partial derivative of z with respect to x at (1, 1, 3) is 3.

Bradley Monk: /* Partial Derivatives */

2013-11-03T22:52:00Z

Partial Derivatives

← Older revision		Revision as of 15:52, 3 November 2013
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	==Partial Derivatives==		==Partial Derivatives==
	In mathematics, a [http://en.wikipedia.org/wiki/Partial_derivative partial derivative] of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.		In mathematics, a [http://en.wikipedia.org/wiki/Partial_derivative partial derivative] of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.


	[[File:Partial Derivative 3D.png]]		[[File:Partial Derivative 3D.png]]
	[[File:Partial Derivative 2D.png]]		[[File:Partial Derivative 2D.png]]


			The partial derivative of a function f with respect to the variable x is variously denoted by:

			[[File:Partial derivative eq1.png]]

	Suppose that ƒ is a function of more than one variable. For instance:		Suppose that ƒ is a function of more than one variable. For instance:

	[[File:Partial derivative ~~eq1~~.png]]		[[File:Partial derivative eq2.png]]

	The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the xz-plane, and those that are parallel to the yz-plane (which result from holding either y or x constant, respectively.)		The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the xz-plane, and those that are parallel to the yz-plane (which result from holding either y or x constant, respectively.)
	To find the slope of the line tangent to the function at P(1, 1, 3) that is parallel to the xz-plane, the y variable is treated as constant. The graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane y = 1. By finding the derivative of the equation while assuming that y is a constant, the slope of ƒ at the point (x, y, z) is found to be:		To find the slope of the line tangent to the function at P(1, 1, 3) that is parallel to the xz-plane, the y variable is treated as constant. The graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane y = 1. By finding the derivative of the equation while assuming that y is a constant, the slope of ƒ at the point (x, y, z) is found to be:

	[[File:Partial derivative ~~eq2~~.png]]		[[File:Partial derivative eq3.png]]

	So at (1, 1, 3), by substitution, the slope is 3. Therefore		So at (1, 1, 3), by substitution, the slope is 3. Therefore

	[[File:Partial derivative ~~eq3~~.png]]		[[File:Partial derivative eq4.png]]

	at the point (1, 1, 3). That is, the partial derivative of z with respect to x at (1, 1, 3) is 3.		at the point (1, 1, 3). That is, the partial derivative of z with respect to x at (1, 1, 3) is 3.

Bradley Monk: /* Gradients */

2013-11-03T22:28:13Z

Gradients

← Older revision		Revision as of 15:28, 3 November 2013
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	==Gradients==		==Gradients==
	[[File:Gradient Math.png]]		[[File:Gradient Math.png\|right]]

	The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, ..., xn) is denoted ∇f or ~~\vec{\~~nabla} f where ∇ (the nabla symbol) denotes the vector differential operator, del. The notation "grad(f)" is also commonly used for the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is,		The [http://en.wikipedia.org/wiki/Gradient gradient] (or gradient vector field) of a scalar function ''f(x1, x2, x3, ..., xn)'' is denoted ∇f or <sup>→</sup>∇f where ∇ (the nabla symbol) denotes the vector differential operator, del. The notation "grad(f)" is also commonly used for the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is,

	[[File:Gradient Eq1.png]]		[[File:Gradient Eq1.png]]

Bradley Monk: /* Laplace Operator */

2013-11-03T22:23:46Z

Laplace Operator

← Older revision		Revision as of 15:23, 3 November 2013
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	==Laplace Operator==		==Laplace Operator==

	~~{{ExpandBox\|Laplace operator\|~~
	In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows.
	~~}}<!-- END ARTICLE -->~~

			In mathematics the [http://en.wikipedia.org/wiki/Laplacian Laplace operator] or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows.

			{{ExpandBox\|more info or media\|
			No additional media
			}}



			==Gradients==
			[[File:Gradient Math.png]]

			The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, ..., xn) is denoted ∇f or \vec{\nabla} f where ∇ (the nabla symbol) denotes the vector differential operator, del. The notation "grad(f)" is also commonly used for the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is,

			[[File:Gradient Eq1.png]]

			In a rectangular coordinate system, the gradient is the vector field whose components are the partial derivatives of f:

			[[File:Gradient Eq2.png]]

			where the ei are the orthogonal unit vectors pointing in the coordinate directions. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.
			In the three-dimensional Cartesian coordinate system, this is given by

			[[File:Gradient Eq3.png]]

			where i, j, k are the standard unit vectors. For example, the gradient of the function

			[[File:Gradient Eq4.png]]

			is:

			[[File:Gradient Eq5.png]]

			In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system.



	[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]]		[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]]

Bradley Monk: /* Partial Derivatives */

2013-11-03T22:15:54Z

Partial Derivatives

← Older revision		Revision as of 15:15, 3 November 2013
Line 29:		Line 29:

	==Partial Derivatives==		==Partial Derivatives==
	In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.		In mathematics, a [http://en.wikipedia.org/wiki/Partial_derivative partial derivative] of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.

	[[File:Partial Derivative 3D.png]]		[[File:Partial Derivative 3D.png]]
Line 48:		Line 48:

	at the point (1, 1, 3). That is, the partial derivative of z with respect to x at (1, 1, 3) is 3.		at the point (1, 1, 3). That is, the partial derivative of z with respect to x at (1, 1, 3) is 3.


	==Laplace Operator==		==Laplace Operator==