Gradient explained with a simple intuitive example!

Gradient term is frequently used in the Electromagnetics. Its sound understanding is needed to reach Maxwell’s equations. This article presents a simple intuitive gradient example for better understanding.

What is the Gradient of a Scalar Field?

In simple words, the Gradient is the other name for multi-variable differentiation in vector fields.

Technically, the Gradient of a scalar field is defined as the magnitude and direction of the maximum space rate of increase of that field.
Space rate indicates the derivative of the function/field with respect to spatial coordinates viz x, y, z etc.
The gradient of the scalar function/field viz. the answer of the gradient operation is a vector which points in the direction of the maximum rate of increase of the given function/field.

For the detailed technical discussion of the Gradient, check out former articles below:-
The gradient of a Scalar Field & The Gradient Operator

The Intuitive Gradient Example

Consider a hypothetical room whose temperature is different at every point and depends on the coordinates of the point. In other words, the temperature at each point inside the room is a function of (x, y, z) coordinates.

We know that temperature is a scalar quantity and it is a function of spatial coordinates. So we can say, the room assumed above is an example of the scalar function or scalar field.

Now, let us assume that this room is like an oven for us and we put a cookie anywhere inside it. But we want to bake the cookie as quickly as possible. Let us also assume that, this imaginary room cum oven posses some magical power by which the object inside it, can float anywhere within the room. In other words, the object can adjust its own position in the space according to the command/ program fed by you.

Is the scene ready in your mind? Now you are a programmer and you are supposed to write a program (of course the simple one) so that the cookie can be baked as quickly as possible, say within the minimum time.

What would you do? Can you think of the flowchart for the required program? What could be your first step? Our aim is to bake the cookie as quickly as possible. How can this be achieved? Any guess?

Of course, this can be achieved by constantly moving the cookie towards the point of higher temperature (than previous) and make it to the point of highest temperature. So our program must always point the cookie towards the neighbourhood point which has a higher temperature than previous. So what our program shoul do? It would check all the direction i.e. x, y and z to find where the change of temperature is maximum. How? Of course, by the derivative. You know the derivative gives us the rate of change of function; in this case, the derivative with respect to x, y and z i.e. spatial coordinates.

Now let us add the corresponding unit vectors with the derivative terms i.e. a_x with X-direction derivative, a_y with Y-direction derivative and a_z with Z-direction. So the combined vector formed is as shown below where T is the temperature inside the room at any point (x, y, z) i.e. the scalar function.

As a whole, this vector will always give the magnitude and the direction of the maximum rate of increase of the temperature inside the room. Can you notice that the vector above is nothing but a Gradient Vector! Isn’t it?

So the program of our Gradient example should actually be finding the Gradient of the scalar field (Temperature inside) and eventually to reach the maximum temperature point.

It should be noted that many facets of the above Gradient Example can be considered for this hypothetical program. Can you comment few?

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