Cartesian Coordinates

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Cartesian coordinates are rectilinear (a special case of curvilinear) two or three dimensional coordinates, with variables labelled x,y, and z. They describe points in a Euclidean Space

When graphing Cartesian Coordinates, the plasmagicians convention, which is also ubiquitous in math and physics, is that of the x coordinate coming out of the diagram, while the z coordinate points upward and y points to the right.

Laplace's equation is separable in Cartesian Coordinates.

Commonly Used Factors and Operators
Value \mathbf{\hat{x}} \mathbf{\hat{y}} \mathbf{\hat{z}}
Scale Factors 1 1 1
Gradient, \mathbf{\nabla} \frac{\partial}{\partial x} \frac{\partial}{\partial y} \frac{\partial}{\partial z}
Laplacian, \nabla^2 \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}
Vector Laplacian, \mathbf{\nabla^2} \mathbf(F) \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \left[{\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}}\right] \mathbf{F}_x \left[{\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}}\right] \mathbf{F}_y \left[{\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}}\right] \mathbf{F}_z
Divergence, \mathbf{\nabla \cdot F} \frac{\partial \mathbf{F}_x}{\partial x} + \frac{\partial \mathbf{F}_y}{\partial y} + \frac{\partial \mathbf{F}_z}{\partial z}
Curl, \mathbf{\nabla \times F} \left|{\begin{bmatrix} \mathbf{\hat{x}} & \mathbf{\hat{y}} & \mathbf{\hat{z}} \\ \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ \\ F_x & F_y & F_z \end{bmatrix}}\right| \frac{\partial \mathbf{F}_z}{\partial y} - \frac{\partial \mathbf{F}_y}{\partial z} \frac{\partial \mathbf{F}_x}{\partial z} - \frac{\partial \mathbf{F}_z}{\partial x} \frac{\partial \mathbf{F}_y}{\partial x} - \frac{\partial \mathbf{F}_x}{\partial y}
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