ODE |
Ordinary Differential Equation
|
PDE |
Partial Differential Equation
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BC |
Boundary Condition
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IVP |
Initial Value Problem
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BVP |
Boundary Value Problem
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IBVP |
Initial Boundary Value Problem
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Operators are shown applied to the scalar
or the vector field
.
Notation |
Common names and other notation |
Description and notes |
Definition in Cartesian coordinates
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![{\displaystyle {\frac {\partial u}{\partial x_{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84313363d3fa39a53fb5741a2075f65c99546ba4) |
Partial derivative, ![{\displaystyle u_{x_{i}},\ \partial _{x_{i}}u\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/016cb9222b3e5f6a09582bf9d62f27372130ba2c) |
The rate of change of with respect to , holding the other independent variables constant. |
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![{\displaystyle {\frac {du}{dx_{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b56d6e7998dec83d3d7a20c4bc6d1fbf31a84e) |
Derivative, total derivative, ![{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} x_{i}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bea7b0e657415a6dc2c193f5baa9876e1cbfb268) |
The rate of change of with respect to . If is multivariate, this derivative will typically depend on the other variables following a path. |
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![{\displaystyle \nabla u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ece25ca70b649b540020fc649944c6f940d302) |
Gradient, del operator, ![{\displaystyle \mathrm {grad} \ u\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27254ffcfb99843fba5f1520efad3ec2a5868ea5) |
Vector that describes the direction and magnitude of the greatest rate of change of a function of more than one variable. The symbol is called nabla. |
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![{\displaystyle \nabla ^{2}u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e525a67d9fa7d7cc07fadbb893236c02c9b670) |
Laplacian, Scalar Laplacian, Laplace operator, ![{\displaystyle \Delta u,\ (\nabla \cdot \nabla )u\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/206c6e876523d5b34adc93610194c68e2e2c9d81) |
A measure of the concavity of , equivalently a comparison of the value of at some point to neighboring values. |
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![{\displaystyle \nabla \cdot \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4fee0cfc620e59850feaf72a21ea92720ff6ac) |
Divergence, ![{\displaystyle \mathrm {div} \ \mathbf {v} \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/061c21961e470fb25e8dc3c28dd7203ac979bc27) |
A measure of "generation", in other words how much the vector field acts as a source or sink at a point. |
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![{\displaystyle \nabla \times \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/07734a9e93cbd0c540364d8f4173b3a0cda23287) |
Curl, rotor, circulation density, ![{\displaystyle \mathrm {curl} \ \mathbf {v} ,\ \mathrm {rot} \ \mathbf {v} \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fc0e3b908132860c80e7af2ccd35bcf542d275) |
A vector that describes the rate of rotation of a (normally 3D) vector field and the corresponding axis of rotation. |
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![{\displaystyle \nabla ^{2}\mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/eac677345fc11377e336e9215264e576b7564365) |
Vector Laplacian |
Similar to the (scalar) Laplacian. Note however, that it is generally not equal to the element-by-element Laplacian of a vector. |
|
Cartesian representations appear in the table above. The
convention is used for spherical coordinates.
Operator |
Cylindrical |
Spherical
|
![{\displaystyle \nabla u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ece25ca70b649b540020fc649944c6f940d302) |
![{\displaystyle \left({\frac {\partial u}{\partial r}},{\frac {1}{r}}{\frac {\partial u}{\partial \theta }},{\frac {\partial u}{\partial z}}\right)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71b987feca6d47ba56a838bf527fbd3a183647c0) |
|
![{\displaystyle \nabla ^{2}u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e525a67d9fa7d7cc07fadbb893236c02c9b670) |
![{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial u}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}u}{\partial \theta ^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dd39407785fc071c47280e9e0f6e4ba7b5c873) |
|
![{\displaystyle \nabla \cdot \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4fee0cfc620e59850feaf72a21ea92720ff6ac) |
![{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rv_{r}\right)+{\frac {1}{r}}{\frac {\partial v_{\theta }}{\partial \theta }}+{\frac {\partial v_{z}}{\partial z}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e82d383dbf4ad2537103d3555302f68246e2fb0d) |
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