# Struggling immensely with tensors in multivariable calculus

In summary: I'm sorry, I'm not quite following what you are saying.I'm afraid even this basic a grasp of tensors is lost to me. What exactly is a tensor in the first place? I don't think I can answer your initial question concerning the fundamental definition of a rank two tensor.What happens to the components of a position vector (tensor of rank 1) ##x_i## (where ##x_1 = x##, ##x_2 = y##, ##x_3 = z##) when you rotate the xyz coordinate system? What about a two index object ##a_{ij} = x_i x_j##, (for which ##a_{11} = x^2

## Homework Statement

If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2.

N/A

## The Attempt at a Solution

I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is supposed to be a part of our curriculum) and this question came up. I'm not sure at all how to answer it. Any help in understanding this problem would be hugely appreciated, even if it doesn't solve the entire thing.

Thank you.

## Homework Statement

If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2.

N/A

## The Attempt at a Solution

I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is supposed to be a part of our curriculum) and this question came up. I'm not sure at all how to answer it. Any help in understanding this problem would be hugely appreciated, even if it doesn't solve the entire thing.

Thank you.
What is the fundamental definition of a Cartesian tensor of rank 2? (Hint: it has to do with what happens when we make a change of coordinates)

nrqed said:
What is the fundamental definition of a Cartesian tensor of rank 2? (Hint: it has to do with what happens when we make a change of coordinates)

Is it a matrix? I've learned that a tensor of rank 0 is a scalar and that a tensor of rank 1 is a vector, but I'm not sure I fully understand what exactly tensors are supposed to represent.

Is it a matrix? I've learned that a tensor of rank 0 is a scalar and that a tensor of rank 1 is a vector, but I'm not sure I fully understand what exactly tensors are supposed to represent.
well, one can think of it as a matrix since there are two indices but this is not important here, what is important is how it transforms under a change of coordinates.

nrqed said:
well, one can think of it as a matrix since there are two indices but this is not important here, what is important is how it transforms under a change of coordinates.

I'm afraid even this basic a grasp of tensors is lost to me. What exactly is a tensor in the first place? I don't think I can answer your initial question concerning the fundamental definition of a rank two tensor.

What happens to the components of a position vector (tensor of rank 1) ##x_i## (where ##x_1 = x##, ##x_2 = y##, ##x_3 = z##) when you rotate the xyz coordinate system? What about a two index object ##a_{ij} = x_i x_j##, (for which ##a_{11} = x^2## and ##a_{12} = xy##, etc.)? Now you should use the chain rule of partial derivatives to show that under rotations the object ##\frac{\partial}{\partial x_i \partial x_j}## has a transformation law that is similar to that of the ##a_{ij}##...

## 1. What are tensors and how are they used in multivariable calculus?

Tensors are mathematical objects that represent linear relationships between different sets of variables. In multivariable calculus, tensors are used to describe the relationships between multiple variables in a system, such as position, velocity, and acceleration.

## 2. Why do students struggle with tensors in multivariable calculus?

Tensors can be difficult to visualize and understand because they involve multiple dimensions and can be represented in different ways. Additionally, the notation used for tensors can be confusing for students who are not familiar with it.

## 3. How can I improve my understanding of tensors in multivariable calculus?

Practice is key when it comes to understanding tensors in multivariable calculus. It is important to work through problems and exercises to become familiar with the notation and concepts. Seeking help from a tutor or professor can also be beneficial.

## 4. What are some real-world applications of tensors in multivariable calculus?

Tensors have many practical applications in fields such as physics, engineering, and computer science. They are used to model and analyze systems with multiple variables, such as fluid flow, stress and strain in materials, and image processing.

## 5. Are there any resources that can help me better understand tensors in multivariable calculus?

There are many online resources, textbooks, and videos available that can help improve understanding of tensors in multivariable calculus. It may also be helpful to seek out a study group or attend extra help sessions with a professor.

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