Gradient

Mathematical notation is something that, often, complicates the formulas we see, mainly because according to who does that is the one that uses.

Come on, that the notation of things is not unique and besides, there are many symbols that do not differ much between them, which gives errors.

Today I’m going to talk about the gradient. The gradient is nothing more than the directional derivative of a scalar field. That is, if we have a scalar field and derive in a given direction the result is the variation of how that field varies in that direction, the gradient.

If we have a scalar field \ varphi (\overrightarrow {r}) and derive in the direction of the vector \ verrightarrow {v}.

\frac{d\varphi }{ds}=\lim_{\Delta s\rightarrow 0}\frac{\varphi(\overrightarrow{r} + \Delta s\overrightarrow{v})-\varphi (\overrightarrow{r})}{\Delta s}

If we put \overrightarrow{v} = (x, y, z) we will have the previous limit (and therefore the derivative since latex \Delta s\rightarrow 0$) is equal to the partial derivatives in each of the And if, in addition, obviously also, we use the rule of the chain we will have the definition of gradient.

\overrightarrow{grad} \varphi = \bigtriangledown \varphi=\frac{\partial \varphi}{\partial x}\overrightarrow{u_{x}}+\frac{\partial \varphi}{\partial y}\overrightarrow{u_{y}}+\frac{\partial \varphi}{\partial z}\overrightarrow{u_{z}}

Where, we point out that while engineers use $ latex \overrightarrow {grad}$, physicists use more \bigtriangledown \varphi. A notation that you will see in many formulas.

That is, we must distinguish $ latex \bigtriangledown$ from \Delta since one is the partial derivative in all components while the other is an increase of a value (which can be a partial derivative in one single component).

As peculiarity is that the gradient is always perpendicular to the surface that has the same value that passes through a point, ie, the gradient is always at 90 degrees between the surface where the scalar field at the point. All of this is because sometimes the gradient is written not in vectorial form (which is the correct thing) but in a scalar way using its module.

| \bigtriangledown \varphi |=| \overrightarrow{\varphi}| cos( \alpha )

Where cos(\alpha) Is the angle that forms the surface with the vector \overrightarrow v which we were talking about at the beginning and which are the chosen coordinate system. Simple.

Sometimes the gradient is also defined as:

\bigtriangledown \varphi (\overrightarrow{r})= \lim_{\bigtriangleup s\rightarrow 0}\frac{1}{\bigtriangleup s}\oint \varphi (\overrightarrow{r})d\overrightarrow{S}(\overrightarrow{r})

That, at bottom, something that sounds so complicated is to indicate that it is the limit (hence the partial) of the calculation of the amount of scalar field that passes through a surface “without definite form” (hence the integral is closed and therefore, for the surface (and the differential is of the surface, vectorially speaking). Nothing in itself that we have not already seen except that instead of for a point, it is for a surface.

Remember that the vector \overrightarrow {v} we take it according to a coordinate system and this coordinate system can vary. They do not have to be Cartesian coordinates but of any other type. In this case, the partial derivatives vary and all we have to do (since we will know or have the application that takes us from the new coordinate system to the old Cartesian system) is the chain rule in the partial derivatives … A mesh but nothing complicated.

For example, if we have a cylindrical coordinate system (the image is not mine, so you can criticize it) we will have to:

x=r cos \phi
y = r sen \phi
z=z

And therefore, with the chain rule, we will have to:

\bigtriangledown \varphi=\frac{\partial \varphi}{\partial r}\overrightarrow{u_{r}}+\frac{1}{r}\frac{\partial \varphi}{\partial \phi }\overrightarrow{u_{\phi }}+\frac{\partial \varphi}{\partial z}\overrightarrow{u_{z}}

The important thing of all this is to know perfectly the mathematical notation that, depending on the field in which we move, varies a little. This will help us to understand better (and not by words) what is meant at all times.

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