The symbol \nabla (nabla) represents the del operator.

Usage

Definition

=(x1,,xn)\nabla = \nail{\fracp{}{x_1}, \dots, \fracp{}{x_n}}

For example, using the standard basis i^,j^,k^\hat i, \hat j, \hat k of R3\RR^3, we get =(x,y,z)=i^x+j^y+k^z\nabla = \nail{\fracp{}{x}, \fracp{}{y}, \fracp{}{z}} = \hat i\fracp{}{x} + \hat j\fracp{}{y} + \hat k\fracp{}{z}

Gradient

If f:R3Rf:\RR^3\to\RR is a scalar field, gradf=f=(fx,fy,fz)R3\Mr{grad} f = \nabla f = \nail{\fracp{f}{x}, \fracp{f}{y}, \fracp{f}{z}} \in \RR^3 The gradient is the "slope" direction and magnitude.

Divergence

If f:R3R3f:\RR^3\to\RR^3 is a vector field and f(x,y,z)=(fx,fy,fz)f(x,y,z) = (f_x,f_y,f_z), divf=f=fxx+fyy+fzzR\Mr{div} f = \nabla\cdot f = \fracp{f_x}{x} + \fracp{f_y}{y} + \fracp{f_z}{z} \in \RR The divergence measures how much field diverges from the given point.

Curl

If f:R3R3f:\RR^3\to\RR^3 is a vector field and f(x,y,z)=(fx,fy,fz)f(x,y,z) = (f_x,f_y,f_z), curlf=×f=xyzfxfyfzi^j^k^R3\Mr{curl} f = \nabla\times f = \abs{\begin{matrix} \fracp{}{x} & \fracp{}{y} & \fracp{}{z} \\ f_x & f_y & f_z \\ \hat i & \hat j & \hat k \end{matrix}}\in\RR^3 The curl is the torque at a given point.

Directional Derivative

If f:R3Rf:\RR^3\to\RR is a scalar field and a(x,y,z)=(ax,ay,az)a(x,y,z) = (a_x,a_y,a_z), agradf=(a)f=axfx+ayfy+azfzRa\cdot\Mr{grad} f = (a\cdot\nabla) f = a_x\fracp{f}{x} + a_y\fracp{f}{y} + a_z\fracp{f}{z} \in \RR

Hessian / Laplacian

This one is confusing. ML people use 2\nabla^2 to denote the Hessian matrix. For f:R3Rf:\RR^3\to\RR, 2f=[2x2f2xyf2xzf2yxf2y2f2yzf2zxf2zyf2z2f]R3\nabla^2 f = \matx{ \fracp{^2}{x^2} f & \fracp{^2}{x\partial y} f & \fracp{^2}{x\partial z} f \\ \fracp{^2}{y\partial x} f & \fracp{^2}{y^2} f & \fracp{^2}{y\partial z} f \\ \fracp{^2}{z\partial x} f & \fracp{^2}{z\partial y} f & \fracp{^2}{z^2} f \\ }\in\RR^3

However, in physics, 2\nabla^2 denotes the Laplacian operator Δf=2f=f=2x2f+2y2f+2z2fR\Delta f = \nabla^2 f = \nabla\cdot\nabla f = \fracp{^2}{x^2} f + \fracp{^2}{y^2} f + \fracp{^2}{z^2} f \in \RR

Both operators can also be applied on f:R3Rhigherf:\RR^3\to\RR^\text{higher}, but the results will have more dimensions.

References

Exported: 2021-01-02T23:40:14.258551