# Fundamental Properties of the Gradient of a Scalar Field

The gradient of a Scalar field is a common operation in the Electromagnetics. This article highlights the fundamental Gradient Properties that are unavoidable for the learner.

## The Gradient of a Scalar Field

In the last few articles, I have discussed in detail about the Gradient of a Scalar Field. For Example:- What is the Gradient? How is it represented? Is it a scalar or vector? What are its magnitude and direction? How the Gradient operation can be intuitively understood in practical life? etc. Let me give you the links for easy reference.

### What is the Gradient of a Scalar Field?

* 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. (Click the Title above)*

### How to represent the Gradient in different Coordinate Systems?

* Let the scalar function or field is ψ. Assume that we are required to apply the gradient operation on this ψ. This would be represented as- ∇ψ. (Click the Title above)*

### Gradient explained with a simple intuitive 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. (Click the Title above)*

## Gradient Properties that you must know!

### The Gradient is like the Derivative

The Gradient can be said as the similar operation of multi-variable differentiation in normal calculus but not exactly the same. In normal calculus the derivative gives us the rate of change of one function with respect to certain variable. It is giving information about the slope of the curve. But the Gradient is the special kind bundle of derivatives with respect to space coordinates (like x, y, z) with unified direction.

### The Gradient operation operates on Scalar and gives back Vector

The answer of Gradient operation is a vector. It is evident from the Gradient Operator that each derivative term is associated with the respective unit vector.

### The Gradient vector points towards the maximum space rate change

The magnitude and direction of the Gradient is the maximum rate of change the scalar field with respect to position i.e. spatial coordinates.

Let me make you understand this with a simple example. Consider the simple scalar function, V = x^{2} + y^{2} + z^{2}. We know that this function, V = constant would give us the sphere. Let us consider the two values of the function V_{1} = 4 and V_{2} = 8. Now we can see that the function V is changing w.r.t. spatial coordinates i.e. x, y and z. The Gradient of the function at a point can be intuitively shown as below.

### Gradient Vector is normal to the constant V Surface

The Gradient vector at any point is perpendicular to the constant V surface that is passing through that point. It could be probably the most useful for Electrostatics, amongst all the Gradient Properties. As seen from the above diagram, V_{1} and V_{2} are the V equal to constant surfaces. We can also assume intuitively that the function is gradually changing from 4 to 8. Hypothetically saying, if I am leaving from V_{1} to reach for V_{2} then I have to follow the perpendicular path between these surface to reach quickly. Isn’t it? In technical language, this is so because the change in the function is highest along the normal and hence the Gradient direction.

Can you comment one of the Gradient Properties that is worth mentioning?

*Suggested Community:* – Electromagnetics for GATE and ESE

Tag:Electromagnetism