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Gradient and Differential of a Scalar Field Function

1. The Gradient of a Scalar Field Function \(F\) (denoted by \(\nabla F\)) and its Negative (denoted by \(-\nabla F\)) in any given \(N\) Dimensional Coordinate System are Vector Field Functions in that Coordinate System which are given as
   
   \(\nabla F = \nabla_1 \mathbf{e_1} + \nabla_2 \mathbf{e_2} + \cdots + \nabla_n \mathbf{e_n}\)   ...(1)
   
   \(-\nabla F = -(\nabla_1 \mathbf{e_1} + \nabla_2 \mathbf{e_2} + \cdots + \nabla_n \mathbf{e_n})\)   ...(2)
   
   where
   
   \(\nabla_1, \nabla_2, \cdots, \nabla_n = \) Components of the Gradient Vector Field Function
   
   \(\mathbf{e_1}, \mathbf{e_2}, \cdots, \mathbf{e_n} = \) Covariant Basis Vectors of the given Coordinate System
   
   The Magnitude of the Gradient of \(F\) or its Negative (denoted by \(|\nabla F|\)) gives the Maximum Rate of Change (Increase or Decrease) of \(F\) at any given location of the Coordinate System.
   
   The Unit Vector corresponding to the Gradient of \(F\) (denoted by \({\Large \frac{\nabla F}{|\nabla F|}}\)) gives the Direction of Maximum Rate of Increase of the Scalar Field Function \(F\) at any given location of the Coordinate System.
   
   Similarly, the Unit Vector corresponding to the Negative of the Gradient of \(F\) (denoted by \(-{\Large \frac{\nabla F}{|\nabla F|}}\)) gives the Direction of Maximum Rate of Decrease of the Scalar Field Function \(F\) at any given location of the Coordinate System.
2. The Differential Change in a Scalar Field Function \(F\) (also called the Differential of the Scalar Field Function \(F\)) denoted by \(dF\) in any given Coordinate System can be calculated as a Dot Product between the Gradient of Function \(F\) (denoted by \(\nabla F\)) and the Differential Displacement Vector of the Coordinate System (denoted by \(d\vec{R}\)) as follows
   
   \(dF=\nabla F \cdot d\vec{R}\)   ...(3)
3. Using the formula of the Differential of the Scalar Field Function \(F\) given in equation (3) above, the Components of Gradient of the Scalar Field Function \(F\) (i.e. \(\nabla F\)) given by \(\nabla_1, \nabla_2, \cdots, \nabla_n \) in any \(N\) Dimensional Coordinate System having Coordinate Variables \(x^1, x^2, \cdots, x^n\) and Covariant Basis Vectors \(\mathbf{e_1}, \mathbf{e_2}, \cdots, \mathbf{e_n}\) can be derived as follows
   
   We know that the Differential of the Scalar Field Function \(F\) denoted by \(dF\) is given as Sum of Partial Differentials of the Function \(F\) with respect to all the Coordinate Variables of the Coordinate System as follows
   
   \(dF = {\Large \frac{\partial F}{\partial x^1}}dx^1 + {\Large \frac{\partial F}{\partial x^2}}dx^2 + \cdots + {\Large \frac{\partial F}{\partial x^n}}dx^n= \begin{bmatrix}{\Large \frac{\partial F}{\partial x^1}} & {\Large \frac{\partial F}{\partial x^2}} & \cdots & {\Large \frac{\partial F}{\partial x^n}}\end{bmatrix} \begin{bmatrix}dx^1 \\ dx^2 \\ \vdots \\ dx^n\end{bmatrix}\)   ...(4)
   
   Also, from equation (1) above we have
   
   \(\nabla F = \nabla_1 \mathbf{e_1} + \nabla_2 \mathbf{e_2} + \cdots + \nabla_n \mathbf{e_n} = \begin{bmatrix}\nabla_1 & \nabla_2 & \cdots & \nabla_n\end{bmatrix} \begin{bmatrix}\mathbf{e_1} \\ \mathbf{e_2} \\ \vdots \\ \mathbf{e_n}\end{bmatrix} \)   ...(5)
   
   And, the Differential Displacement Vector of the Coordinate System \(d\vec{R}\) is given as
   
   \(d\vec{R} = dx^1\mathbf{e_1} + dx^2\mathbf{e_2} + \cdots + dx^n\mathbf{e_n} = \begin{bmatrix}\mathbf{e_1} & \mathbf{e_2} & \cdots & \mathbf{e_n}\end{bmatrix} \begin{bmatrix}dx^1 \\ dx^2 \\ \vdots \\ dx^n\end{bmatrix}\)   ...(6)
   
   Now, Substituting the value of \(dF\), \(\nabla F\) and \(d\vec{R}\) given in equations (4), (5) and (6) respectively in equation (3) above we get
   
   \({\Large \frac{\partial F}{\partial x^1}}dx^1 + {\Large \frac{\partial F}{\partial x^2}}dx^2 + \cdots + {\Large \frac{\partial F}{\partial x^n}}dx^n =(\nabla_1 \mathbf{e_1} + \nabla_2 \mathbf{e_2} + \cdots + \nabla_n \mathbf{e_n}) \cdot (dx^1\mathbf{e_1} + dx^2\mathbf{e_2} + \cdots + dx^n\mathbf{e_n})\)
   
   \(\Rightarrow \begin{bmatrix}{\Large \frac{\partial F}{\partial x^1}} & {\Large \frac{\partial F}{\partial x^2}} & \cdots & {\Large \frac{\partial F}{\partial x^n}}\end{bmatrix} \begin{bmatrix}dx^1 \\ dx^2 \\ \vdots \\ dx^n\end{bmatrix}= \begin{bmatrix}\nabla_1 & \nabla_2 & \cdots & \nabla_n\end{bmatrix} \begin{bmatrix}\mathbf{e_1} \\ \mathbf{e_2} \\ \vdots \\ \mathbf{e_n}\end{bmatrix} \begin{bmatrix}\mathbf{e_1} & \mathbf{e_2} & \cdots & \mathbf{e_n}\end{bmatrix} \begin{bmatrix}dx^1 \\ dx^2 \\ \vdots \\ dx^n\end{bmatrix}\)
   
   \(\Rightarrow \begin{bmatrix}{\Large \frac{\partial F}{\partial x^1}} & {\Large \frac{\partial F}{\partial x^2}} & \cdots & {\Large \frac{\partial F}{\partial x^n}}\end{bmatrix} \begin{bmatrix}dx^1 \\ dx^2 \\ \vdots \\ dx^n\end{bmatrix}= \begin{bmatrix}\nabla_1 & \nabla_2 & \cdots & \nabla_n\end{bmatrix} \begin{bmatrix}\mathbf{e_1} \cdot \mathbf{e_1} & \mathbf{e_1} \cdot \mathbf{e_2} & \cdots & \mathbf{e_1} \cdot \mathbf{e_n} \\ \mathbf{e_2} \cdot \mathbf{e_1} & \mathbf{e_2} \cdot \mathbf{e_2} & \cdots & \mathbf{e_2} \cdot \mathbf{e_n} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{e_n} \cdot \mathbf{e_1} & \mathbf{e_n} \cdot \mathbf{e_2} & \cdots & \mathbf{e_n} \cdot \mathbf{e_n} \end{bmatrix} \begin{bmatrix}dx^1 \\ dx^2 \\ \vdots \\ dx^n\end{bmatrix}\)
   
   \(\Rightarrow \begin{bmatrix}{\Large \frac{\partial F}{\partial x^1}} & {\Large \frac{\partial F}{\partial x^2}} & \cdots & {\Large \frac{\partial F}{\partial x^n}}\end{bmatrix} = \begin{bmatrix}\nabla_1 & \nabla_2 & \cdots & \nabla_n\end{bmatrix} \begin{bmatrix}\mathbf{e_1} \cdot \mathbf{e_1} & \mathbf{e_1} \cdot \mathbf{e_2} & \cdots & \mathbf{e_1} \cdot \mathbf{e_n} \\ \mathbf{e_2} \cdot \mathbf{e_1} & \mathbf{e_2} \cdot \mathbf{e_2} & \cdots & \mathbf{e_2} \cdot \mathbf{e_n} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{e_n} \cdot \mathbf{e_1} & \mathbf{e_n} \cdot \mathbf{e_2} & \cdots & \mathbf{e_n} \cdot \mathbf{e_n} \end{bmatrix}\)
   
   \(\Rightarrow \begin{bmatrix}\nabla_1 & \nabla_2 & \cdots & \nabla_n\end{bmatrix} = \begin{bmatrix}{\Large \frac{\partial F}{\partial x^1}} & {\Large \frac{\partial F}{\partial x^2}} & \cdots & {\Large \frac{\partial F}{\partial x^n}}\end{bmatrix} \begin{bmatrix}\mathbf{e_1} \cdot \mathbf{e_1} & \mathbf{e_1} \cdot \mathbf{e_2} & \cdots & \mathbf{e_1} \cdot \mathbf{e_n} \\ \mathbf{e_2} \cdot \mathbf{e_1} & \mathbf{e_2} \cdot \mathbf{e_2} & \cdots & \mathbf{e_2} \cdot \mathbf{e_n} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{e_n} \cdot \mathbf{e_1} & \mathbf{e_n} \cdot \mathbf{e_2} & \cdots & \mathbf{e_n} \cdot \mathbf{e_n} \end{bmatrix}^{-1}\)
   
   \(\Rightarrow \begin{bmatrix}\nabla_1 & \nabla_2 & \cdots & \nabla_n\end{bmatrix} = \begin{bmatrix}{\Large \frac{\partial F}{\partial x^1}} & {\Large \frac{\partial F}{\partial x^2}} & \cdots & {\Large \frac{\partial F}{\partial x^n}}\end{bmatrix} \begin{bmatrix}g^{11} & g^{12} & \cdots & g^{1n} \\ g^{21} & g^{22} & \cdots & g^{2n} \\ \vdots & \vdots & \ddots & \vdots \\ g^{n1} & g^{n2} & \cdots & g^{nn} \end{bmatrix}\)   ...(7)
   
   where \(\begin{bmatrix}\mathbf{e_1} \cdot \mathbf{e_1} & \mathbf{e_1} \cdot \mathbf{e_2} & \cdots & \mathbf{e_1} \cdot \mathbf{e_n} \\ \mathbf{e_2} \cdot \mathbf{e_1} & \mathbf{e_2} \cdot \mathbf{e_2} & \cdots & \mathbf{e_2} \cdot \mathbf{e_n} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{e_n} \cdot \mathbf{e_1} & \mathbf{e_n} \cdot \mathbf{e_2} & \cdots & \mathbf{e_n} \cdot \mathbf{e_n} \end{bmatrix}^{-1}=\begin{bmatrix}g^{11} & g^{12} & \cdots & g^{1n} \\ g^{21} & g^{22} & \cdots & g^{2n} \\ \vdots & \vdots & \ddots & \vdots \\ g^{n1} & g^{n2} & \cdots & g^{nn} \end{bmatrix}\) is the Inverse Metric Tensor of the Coordinate System
   
   From equations (5) and (7) we get
   
   \(\nabla F = \begin{bmatrix}\nabla_1 & \nabla_2 & \cdots & \nabla_n\end{bmatrix} \begin{bmatrix}\mathbf{e_1} \\ \mathbf{e_2} \\ \vdots \\ \mathbf{e_n}\end{bmatrix}= \begin{bmatrix}{\Large \frac{\partial F}{\partial x^1}} & {\Large \frac{\partial F}{\partial x^2}} & \cdots & {\Large \frac{\partial F}{\partial x^n}}\end{bmatrix} \begin{bmatrix}g^{11} & g^{12} & \cdots & g^{1n} \\ g^{21} & g^{22} & \cdots & g^{2n} \\ \vdots & \vdots & \ddots & \vdots \\ g^{n1} & g^{n2} & \cdots & g^{nn} \end{bmatrix}\begin{bmatrix}\mathbf{e_1} \\ \mathbf{e_2} \\ \vdots \\ \mathbf{e_n}\end{bmatrix}\)   ...(8)