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Definition/Summary
The gradient is a vector operator denoted by the symbol \mathbf\nabla or grad. The gradient of a differentiable scalar function f\left({\mathbf x}\right) of a vector \mathbf{x}=\left(x_1,x_2,\ldots,x_n\right) is a vector field whose components are the partial derivatives of f\left({\mathbf x}\right) with respect to the variables x_1,x_2,\ldots,x_n\,. Explicitly,
\mathbf\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n}\right)
Equations
For a function of three variables in Cartesian Coordinates,
\nabla f\left(x,y,z\right) <br /> = \frac{\partial f}{\partial x}\hat{\mathbf{i}} <br /> + \frac{\partial f}{\partial y}\hat{\mathbf{j}} <br /> + \frac{\partial f}{\partial z}\hat{\mathbf{k}}
In Cylindrical Polar Coordinates,
\nabla f\left(r,\theta,z\right) <br /> = \frac{\partial f}{\partial r}\hat{\mathbf{e_r}} <br /> + \frac{1}{r}\frac{\partial f}{\partial \theta}\hat{\mathbf{e_\theta}} <br /> + \frac{\partial f}{\partial z}\hat{\mathbf{k}}
Where \hat{\mathbf{e_r}} and \hat{\mathbf{e_\theta}} are unit vectors in the radial and angular directions respectively.
In spherical coordinates,
\nabla f\left(r,\phi,\theta\right) <br /> = \frac{\partial f}{\partial r}\hat{\mathbf{e_r}}<br /> + \frac{1}{r} \ \frac{\partial f}{\partial \phi} \hat{\mathbf{e_\phi}}<br /> + \frac{1}{r \ \sin \phi} \ \frac{\partial f}{\partial \theta} \hat{\mathbf{e_\theta}}
where \phi is the angle from the +z-axis to the point (r, \phi, \theta ). Also \hat{\mathbf{e_r}}, etc., denote unit vectors.
NOTE: this definition of \phi, \theta is the one commonly used in math and engineering textbooks. PHYSICS TEXTBOOKS USUALLY HAVE \phi, \theta DEFINED THE OTHER WAY ROUND.
Extended explanation
The main property of the gradient of f, is that it lies in the domain of the function f, and points in the direction in which f is increasing fastest. In particular the gradient at a point \mathbf{p} is perpendicular to the "level set" of f through \mathbf{p}, where f is constantly equal to f(\mathbf{p}).
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
The gradient is a vector operator denoted by the symbol \mathbf\nabla or grad. The gradient of a differentiable scalar function f\left({\mathbf x}\right) of a vector \mathbf{x}=\left(x_1,x_2,\ldots,x_n\right) is a vector field whose components are the partial derivatives of f\left({\mathbf x}\right) with respect to the variables x_1,x_2,\ldots,x_n\,. Explicitly,
\mathbf\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n}\right)
Equations
For a function of three variables in Cartesian Coordinates,
\nabla f\left(x,y,z\right) <br /> = \frac{\partial f}{\partial x}\hat{\mathbf{i}} <br /> + \frac{\partial f}{\partial y}\hat{\mathbf{j}} <br /> + \frac{\partial f}{\partial z}\hat{\mathbf{k}}
In Cylindrical Polar Coordinates,
\nabla f\left(r,\theta,z\right) <br /> = \frac{\partial f}{\partial r}\hat{\mathbf{e_r}} <br /> + \frac{1}{r}\frac{\partial f}{\partial \theta}\hat{\mathbf{e_\theta}} <br /> + \frac{\partial f}{\partial z}\hat{\mathbf{k}}
Where \hat{\mathbf{e_r}} and \hat{\mathbf{e_\theta}} are unit vectors in the radial and angular directions respectively.
In spherical coordinates,
\nabla f\left(r,\phi,\theta\right) <br /> = \frac{\partial f}{\partial r}\hat{\mathbf{e_r}}<br /> + \frac{1}{r} \ \frac{\partial f}{\partial \phi} \hat{\mathbf{e_\phi}}<br /> + \frac{1}{r \ \sin \phi} \ \frac{\partial f}{\partial \theta} \hat{\mathbf{e_\theta}}
where \phi is the angle from the +z-axis to the point (r, \phi, \theta ). Also \hat{\mathbf{e_r}}, etc., denote unit vectors.
NOTE: this definition of \phi, \theta is the one commonly used in math and engineering textbooks. PHYSICS TEXTBOOKS USUALLY HAVE \phi, \theta DEFINED THE OTHER WAY ROUND.
Extended explanation
The main property of the gradient of f, is that it lies in the domain of the function f, and points in the direction in which f is increasing fastest. In particular the gradient at a point \mathbf{p} is perpendicular to the "level set" of f through \mathbf{p}, where f is constantly equal to f(\mathbf{p}).
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!