Struggling immensely with tensors in multivariable calculus

[ParabolaDog]

Homework Statement

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

Homework Equations

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.

 

[nrqed]

Homework Statement

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

Homework Equations

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)

 

[ParabolaDog]

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.

 

[nrqed]

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.

 

[ParabolaDog]

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.

 

[hilbert2]

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 $a_{ij}$...