# Textbook gives the gradient of a scalar as a scalar

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• ENgez
In summary, the conversation discusses homogenization theory and a simple conductivity model with a scalar function, u. The textbook presents a series expansion for the gradient of u, but it is a scalar instead of a vector, which makes the divergence operation undefined. The conversation then questions how to make sense of this and mentions difficulty in referencing equations. However, the speaker believes they have found the cause of their confusion - the model problem is 1-dimensional, making the gradient and divergence essentially the same thing. This resolves the issue and allows for the math to work out.
ENgez
Background: I am currently reading up on homogenization theory.

I have a simple conductivity model (image attached). u is a scalar function (such as potential or temperature).

The textbook proceeds by giving a series expansion for the gradient of u (image attached). the problem is that the gradient is supposed to be a vector function, and the one given is a scalar (pretty sure that the expression given by the book is the differential)

The fact that the gradient is supposedly a scalar makes the divergence operation in the model undefined, which basically makes all following math in the book unclear to me..

How to makes sense of it?

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Pity we can't read the numbers of the equations; this makes referring to them confusing.

'Thiis series is plugged into the equation' -- study 'the' equation to find out what they mean with ##\nabla_y## and ##\nabla _x## --- my guess is that 's a scalar equation, e.g. for the ##z## component, not a vector equation.

But only a guess, given the little I have to go on.

Thanks, I believe I have found the cause of my confusion.

The model problem is 1-dimensional (x is the only free parameter). This makes the gradient a scalar in the sense that it is a magnitude that can either be positive or negative.
This also makes the divergence and the gradient basically mean the same thing.

This seems to make things work out..

## 1. What is the gradient of a scalar?

The gradient of a scalar is a vector that represents the rate of change or slope of a scalar quantity in a particular direction. It is calculated by taking the partial derivatives of the scalar with respect to each independent variable.

## 2. How is the gradient of a scalar represented in a textbook?

In a textbook, the gradient of a scalar is typically represented as the symbol ∇ followed by the scalar variable. For example, the gradient of a scalar quantity f(x,y) would be written as ∇f(x,y).

## 3. What is the difference between a scalar and a vector?

A scalar is a quantity that only has magnitude, while a vector has both magnitude and direction. The gradient of a scalar is a vector because it represents both the magnitude and direction of change of the scalar quantity.

## 4. How is the gradient of a scalar used in science?

The gradient of a scalar is used in many fields of science, including physics, engineering, and mathematics. It is particularly useful in analyzing the behavior of physical systems, such as fluid flow, temperature changes, and electric fields.

## 5. Can the gradient of a scalar be negative?

Yes, the gradient of a scalar can be negative. This indicates that the scalar quantity is decreasing in value in the direction of the gradient. A positive gradient indicates that the scalar is increasing in value in the direction of the gradient.

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