# Understanding dual basis & scale factors

• Incand
In summary, the passage discusses alternative methods for choosing base vectors in curvilinear coordinate systems, including using the normal vector and multiplying it by the scale factors to obtain normalized base vectors. The jacobian is only equal to the product of the scale factors in orthogonal coordinate systems.
Incand
I'm confused by the following passage in our book (translated).
An alternative too choosing the normed tangent vectors ##\vec e_i = \frac{1}{h_1}\frac{\partial \vec r}{\partial u_i}## with scale factors ##h_i = \left| \frac{\partial \vec r }{\partial u_i} \right| ## is to choose the normal vector ##\nabla u_i##.
If the system is ortogonal the vectors ##\nabla u_i## and ##\frac{\partial \vec r}{\partial u_i}## point in the same direction so we can write ##\vec e_i = h_i \nabla u_i##.

##\{ u_i \}_{i=1}^3## is supposed to be curvilinear coordinates with a transformation describing the position ##\vec r = \vec r (u_1,u_2,u_3)##.
What does it mean to take ##\nabla u_i##? am I supposed to express ##u_i## in cartesian coordinates and then take the gradient? And why multiply with the scale factors instead of dividing?
Another question I'm wondering about is if it's always true that the jacobian ##J## is equal too ##h_1h_2h_3##.

Incand said:
##\{ u_i \}_{i=1}^3## is supposed to be curvilinear coordinates with a transformation describing the position ##\vec r = \vec r (u_1,u_2,u_3)##.
What does it mean to take ##\nabla u_i##? am I supposed to express ##u_i## in cartesian coordinates and then take the gradient?

That would be one way of doing it yes. The coordinates ##u_i## are functions and therefore you can take the gradient of them.

Incand said:
And why multiply with the scale factors instead of dividing?

Because this is how you would obtain normalised base vectors. The norm of the dual basis is the reciprocal of the norm of the tangent vector basis.

Incand said:
Another question I'm wondering about is if it's always true that the jacobian ##J## is equal too ##h_1h_2h_3##.
It is only true in orthogonal coordinate systems (which are the only ones for which it makes sense to deal with the scale factors rather than the full metric tensor).

Incand

## 1. What is a dual basis in mathematics?

A dual basis is a set of vectors that can be used to express any vector in a given vector space as a linear combination of those dual basis vectors. It is a fundamental concept in linear algebra and is used in various applications such as solving systems of linear equations and understanding transformation matrices.

## 2. How is a dual basis related to scale factors?

In the context of geometry, scale factors are used to describe how much a shape is enlarged or reduced when it is transformed. In linear algebra, scale factors are represented by diagonal matrices, and the dual basis vectors are used to determine the scale factors for each dimension in a given vector space.

## 3. What is the importance of understanding dual basis and scale factors?

Understanding dual basis and scale factors is crucial in many areas of mathematics and science, as they provide a way to represent and manipulate vectors and transformations. They are also used in the development of computer graphics, machine learning, and other fields that require linear algebra.

## 4. How are dual basis and scale factors calculated?

Dual basis vectors can be calculated using the Gram-Schmidt process, which involves orthogonalizing a set of vectors. Scale factors can be calculated by taking the norm of the dual basis vectors and dividing by the norm of the original basis vectors.

## 5. Can dual basis and scale factors be applied to non-linear transformations?

While dual basis and scale factors are primarily used for linear transformations, they can also be extended to non-linear transformations through the use of differential calculus. In these cases, the scale factors will vary for different points in the vector space.

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