Gradient calculus is frequently used in Electromagnetic Theory. Particularly, it is significant while understanding the relation between Electric Field and Potential. This article discusses the detailed definition of the Gradient in Electromagnetics.
A formal definition of the Gradient
The Gradient of the scalar function/field is a vector representing both the magnitude and direction of the maximum space rate (derivative w.r.t. spatial coordinates) of increase of that function/field.
The Gradient in simple words
Regardless of the fancy definition above, you can assume the “Gradient” as the other name for the “Differentiation”.
In a single variable function, say y = f(x), the derivative (dy/dx) represents the rate of change of ‘y‘ with respect to ‘x‘. In other words how much is the change in ‘y‘ (dy) for a change in ‘x‘ (dx). It represents the slope of the tangent at that point.
In multivariable function, the value of the function may change with respect to any variable, because of change in one or two or all. So it can have three different derivatives for each coordinate. The combined effect with a direction may be considered as the Gradient.
The Gradient – Explained in detail
For simplicity consider the Cartesian coordinate scalar function U (x, y, z). U = constant would represent the certain, surface/contour as shown in the figure for U = U1, U = U2, U = U3.
The small change in this function i.e. dU can be because of the change in x or y or z or all. The change can be thought as the surface U1 tries to occupy U2 so the change is U2 – U1 and likewise.
The change [dU]x because of change in x can be represented as- (∂U/∂x)dx. This is according to the elementary definition of the derivative. As we want the change in U introduced because of change in x, firstly we need the rate of change of U w.r.t. x i.e. (∂U/∂x) then multiplied by the small change dx. The derivative used here is the partial derivative because the function U is also the function of other variables and when we consider the change in it because of x, we inherently assuming that other variable i.e. y and z to be constants.
On a similar line, we can write the [dU]y and [dU]z and finally the change in U i.e. dU can be represented as
dU= (∂U/∂x)dx + (∂U/∂y)dy + (∂U/∂z)dz
Let us slightly modify the above equation as –
Aren’t previous equation and this equation same? 100% same. Try the dot product above and you will end up with the former equation. We have modified the equation for a reason, you come to know.
Let us call the first bracketed term as G and the term in the second bracket, as you already know in the topic of line integration, is dl. So dU can be written as,
So, we can write as,
Carefully examine the term what we called ‘G‘
Can you guess, what does this term represent?
This G is representing the rate of change of function U w.r.t. spatial coordinates x, y and z i.e. the space rate. (Have you recalled this word from the definition?)
One more thing you can easily notice that G is indicating the rate of change of the complete function comprising the x-component i.e. change along x, y-component i.e. change along y and z-component i.e. the change along z. So overall, with vector representation, it is giving us the magnitude as well the direction for the rate of change of U w.r.t. x, y and z. Summarizing G is denoting the magnitude and direction of the space rate of increase of the scalar function U.
Note that, I have used the word “Rate of increase in U“. Can you guess why have I done so? The G is giving us the rate of change of function U, that’s fine. We discussed this well up to this point. But it can be the rate of increase or maybe the rate decrease. Then why I am saying only the rate of increase.
This can be understood as – with the incremental increase in the spatial variables (dx, dy, dz which are taken positive only) if the final value of U is higher than the initial value then the vector direction is towards final value with positive unit vectors. But if its final value is less than the initial value then the direction would be reversed because of the negative sign appearing with the unit vectors. Why negative sign? Apply the simple rule, change equals final minus initial. In other words, this direction is also pointing the higher value.
So summarizing over the complete 3D space, the direction of G would always be pointing towards the higher value of the scalar function U. So we can say that G is denoting the magnitude and direction of the space rate of increase of the scalar function U.
Why am I talking so much about this G?
Because recall the definition for gradient – The Gradient of the scalar function/field is a vector representing both the magnitude and direction of the maximum space rate (derivative w.r.t. spatial coordinates) of increase of that function/field.
From the definition, we can say, the term G is satisfying all the conditions to be called the Gradient operator. Hence the gradient of the scalar field U is given by –
One point is worth noting here that according to the definition, the Gradient is the maximum space rate of the function. Once again have the look at the equation –
dU/dl is the space rate i.e. rate of change of U w.r.t. x, y and z. And for this to be maximum, it would be along the G so θ = 0 as shown in the diagram below.
So we can define the Gradient as,
The Gradient Symbol
The gradient of the scalar field U is denoted as ∇ U.
The direction of a Gradient Vector
So we have discussed the definition for the Gradient. In one sentence, it is magnitude and direction of maximum space rate of a scalar function. Now, can you draw this Gradient vector (G or ∇ U) indicating proper direction?
The direction of the Gradient Vector must be perpendicular or normal to the given scalar field/function. Refer to the diagram just above, the direction of G is normal to the U1. Can you drop the comments, why so?
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