Editing Gradient (section)

===General coordinates===
We consider [[Curvilinear coordinates|general coordinates]], which we write as {{math|''x''<sup>1</sup>, …, ''x''<sup>''i''</sup>, …, ''x''<sup>''n''</sup>}}, where {{mvar|n}} is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so {{math|''x''<sup>2</sup>}} refers to the second component—not the quantity {{math|''x''}} squared. The index variable {{math|''i''}} refers to an arbitrary element {{math|''x''<sup>''i''</sup>}}. Using [[Einstein notation]], the gradient can then be written as:

<math display="block">\nabla f = \frac{\partial f}{\partial x^{i}}g^{ij} \mathbf{e}_j</math> (Note that its [[Dual space|dual]] is <math display="inline">\mathrm{d}f = \frac{\partial f}{\partial x^{i}}\mathbf{e}^i</math>),

where <math>\mathbf{e}^i = \mathrm{d}x^i</math> and <math>\mathbf{e}_i = \partial \mathbf{x}/\partial x^i</math> refer to the unnormalized local [[Curvilinear coordinates#Covariant and contravariant bases|covariant and contravariant bases]] respectively, <math>g^{ij}</math> is the [[Metric tensor#Inverse metric|inverse metric tensor]], and the Einstein summation convention implies summation over ''i''  and ''j''.

If the coordinates are orthogonal we can easily express the gradient (and the [[Differential form|differential]]) in terms of the normalized bases, which we refer to as  <math>\hat{\mathbf{e}}_i</math> and  <math>\hat{\mathbf{e}}^i</math>, using the scale factors (also known as [[Lamé coefficients]])  <math>h_i= \lVert \mathbf{e}_i \rVert = \sqrt{g_{i i}} = 1\, / \lVert \mathbf{e}^i \rVert</math> :

<math display="block">\nabla f = \frac{\partial f}{\partial x^{i}}g^{ij} \hat{\mathbf{e}}_{j}\sqrt{g_{jj}} = \sum_{i=1}^n \, \frac{\partial f}{\partial x^{i}} \frac{1}{h_i} \mathbf{\hat{e}}_i</math> (and <math display="inline">\mathrm{d}f = \sum_{i=1}^n \, \frac{\partial f}{\partial x^{i}} \frac{1}{h_i} \mathbf{\hat{e}}^i</math>),

where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices, <math>\mathbf{\hat{e}}_i</math>, <math>\mathbf{\hat{e}}^i</math>, and <math>h_i</math> are neither contravariant nor covariant.

The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.