Notational problem in tensor calculus

• jacobrhcp
In summary, The conversation is about using the Einstein convention to calculate explicitly the partial derivatives of \nabla \bullet (x_{i}\vec{a}). The notation x_{i}\vec{a} is clarified and it is determined that the answer is \partial_{j} (a_{j}x_{i}), which simplifies to a_j \partial_{j} x_{i}.
jacobrhcp
Using the Einstein convention, is this about right? (indexes run from 1 to 3):

$$\nabla\bullet(x_{i}a)=div(x_{i}a)=\partial_{j}a_{j}x_{i}=3x_{i}$$

The partial derivatives are take wrt what variable ? x or a ?

it's said x above the exercises $$\partial_{j}=\frac{\partial}{\partial x_{j}}}$$

and the question is:

calculate explicitly: $$\nabla \bullet (x_{i}\vec{a})$$, where a is a constant vector...

my attempt is way off, but I don't feel at home in these new symbols enough to get the right answer.

Last edited:
$$x_i{\vec a}$$ is not clear notation. Does it mean the i component of a tensor
[xa], or does it mean $${\vec\hat i}\cdot{\vec r}{\vec a}$$?

jacobrhcp said:
Using the Einstein convention, is this about right? (indexes run from 1 to 3):

$$\nabla\bullet(x_{i}a)=div(x_{i}a)=\partial_{j}a_{j}x_{i}=3x_{i}$$

You are right that it means $$\partial_{j} (a_{j}x_{i})$$ But the components of a are constants so this is equal to $$a_j \partial_{j} x_{i}$$ Now, what does $$\partial_{j} x_{i}$$give ?

1. What is a tensor in calculus?

A tensor in calculus is a mathematical object used to describe relationships between different quantities in multi-dimensional space. It is represented by an array of numbers and has certain properties that allow it to be manipulated algebraically.

2. What is the notation used to represent tensors in calculus?

Tensors in calculus are typically represented using index notation, where indices are used to denote the position of each element in the tensor array. The number of indices corresponds to the number of dimensions in the space being described.

3. Why is there a notational problem in tensor calculus?

The notation used to represent tensors in calculus can be complex and difficult to understand, leading to confusion and potential errors in calculations. Additionally, there are multiple notations used by different fields of study, making it challenging for scientists to communicate and collaborate.

4. How can the notational problem in tensor calculus be addressed?

One way to address the notational problem in tensor calculus is to use a standardized notation that is widely accepted by the scientific community. This can help improve communication and understanding between researchers. Additionally, there are efforts to develop more intuitive notations for tensors, such as graphical representations.

5. What are some common mistakes made when working with tensors in calculus?

Some common mistakes when working with tensors in calculus include forgetting to include all necessary indices, incorrectly contracting or multiplying tensors, and not understanding the properties of tensors. It is important to carefully follow the rules and properties of tensors to avoid errors in calculations.

Replies
6
Views
1K
Replies
5
Views
2K
Replies
7
Views
1K
Replies
9
Views
1K
Replies
7
Views
1K
Replies
1
Views
826
Replies
30
Views
5K
Replies
3
Views
1K
Replies
6
Views
3K
Replies
1
Views
2K