basic_tools:vector_calculus:gradient
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basic_tools:vector_calculus:gradient [2017/12/16 14:45] jakobadmin [Why is it interesting?] |
basic_tools:vector_calculus:gradient [2018/03/28 12:25] (current) jakobadmin |
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| ====== Gradient ====== | ====== Gradient ====== | ||
| - | <tabbox Why is it interesting?> | + | <tabbox Intuitive> |
| - | The gradient, denoted $\nabla$ (spoken "Nabla"), is a tool that enables us to calculate how much a given function changes in different directions. A "normal" function $f(x)$, lives in a boring one-dimensional space, and the derivative is just another function: $\partial_x f(x)$. This function is the rate of change of $f(x)$. Our real world is three-dimensional and hence in physics we often encounter functions that depend on all spatial directions: $f(x,y,z)$. The gradient of such a function: | + | |
| - | + | ||
| - | $$\nabla f(x,y,z) = \begin{pmatrix} \partial_x f(x,y,z) \\ \partial_x f(x,y,z)\\ \partial_x f(x,y,z) \end{pmatrix}$$ | + | |
| - | + | ||
| - | is a vector and tells us how much $f(x,y,z)$ changes in each direction. | + | |
| - | + | ||
| - | The gradient is especially important in [[theories:classical_theories:electrodynamics|electrodynamics]], because the fundamental equations of electrodynamics (called [[equations:maxwell_equations|Maxwell equations]]) contain the curl of the electric and magnetic fields. | + | |
| - | <tabbox Layman> | + | |
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| - | <tabbox Student> | + | <tabbox Concrete> |
| * [[https://betterexplained.com/articles/vector-calculus-understanding-the-gradient/|Vector Calculus: Understanding the Gradient]] by Kalid Azad | * [[https://betterexplained.com/articles/vector-calculus-understanding-the-gradient/|Vector Calculus: Understanding the Gradient]] by Kalid Azad | ||
| * [[https://betterexplained.com/articles/understanding-pythagorean-distance-and-the-gradient/|Understanding Pythagorean Distance and the Gradient]] by Kalid Azad | * [[https://betterexplained.com/articles/understanding-pythagorean-distance-and-the-gradient/|Understanding Pythagorean Distance and the Gradient]] by Kalid Azad | ||
| - | <tabbox Researcher> | + | <tabbox Abstract> |
| <note tip> | <note tip> | ||
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| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| + | The gradient, denoted $\nabla$ (spoken "Nabla"), is a tool that enables us to calculate how much a given function changes in different directions. A "normal" function $f(x)$, lives in a boring one-dimensional space, and the derivative is just another function: $\partial_x f(x)$. This function is the rate of change of $f(x)$. Our real world is three-dimensional and hence in physics we often encounter functions that depend on all spatial directions: $f(x,y,z)$. The gradient of such a function: | ||
| - | --> Example1# | + | $$\nabla f(x,y,z) = \begin{pmatrix} \partial_x f(x,y,z) \\ \partial_y f(x,y,z)\\ \partial_z f(x,y,z) \end{pmatrix}$$ |
| - | + | is a vector and tells us how much $f(x,y,z)$ changes in each direction. | |
| - | <-- | + | |
| - | + | ||
| - | --> Example2:# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| <tabbox FAQ> | <tabbox FAQ> | ||
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| See http://mathinsight.org/gradient_vector | See http://mathinsight.org/gradient_vector | ||
| | | ||
| - | <tabbox History> | + | <-- |
| </tabbox> | </tabbox> | ||
basic_tools/vector_calculus/gradient.1513431951.txt.gz · Last modified: 2017/12/16 13:45 (external edit)
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