# Meaning of cross terms in line element

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• beefbrisket
In summary, the conversation discusses the line element in non-Cartesian coordinates and the condition for the new coordinate curves to intersect at right angles. The absence of cross terms in the line element indicates that the coordinate lines are orthogonal to each other, as their tangent vectors are orthogonal.
beefbrisket
In a problem from Hartle's Gravity, we are asked to express the line element in non-Cartesian coordinates u, v which are defined with respect to x, y. I have no problem getting the new expression for the line element, but then we are asked if the new coordinate curves intersect at right angles, and the solution says they do because there are no cross terms, du * dv. What is the logic here?

I've taken vector calculus, and try as I might I cannot seem to figure out why the absence of such terms indicates the curves intersect at right angles. I think I might be a little confused since I am interpreting du and dv as scalar values.

The line element is given by ##ds^2 = g_{ij} dx^i dx^j##. Hence, if there are cross terms, then there are off-diagonal entries in the metric. Since the coordinate lines have the holonomic basis vectors ##\partial_i## as their tangent vectors, it follows that the inner product between the tangent vectors of two coordinate lines is given by
$$g(\partial_i,\partial_j) = g_{ij}$$
by definition. Therefore, if ##g_{ij} = 0## for ##i \neq j##, then the coordinate lines are orthogonal to each other since their tangent vectors are orthogonal.

beefbrisket

## 1. What are cross terms in a line element?

Cross terms in a line element refer to the terms that involve more than one variable in the equation. These terms represent the interaction between the different variables and show how they affect each other.

## 2. Why are cross terms important in determining the meaning of a line element?

Cross terms play a crucial role in understanding the relationship between different variables in a system. They help to identify the presence of any interaction or dependence between the variables, which can have a significant impact on the overall behavior of the system.

## 3. How do cross terms affect the overall meaning of a line element?

Cross terms can significantly affect the meaning of a line element by indicating the presence of interaction and dependence between variables. They also provide information about the direction and strength of the relationship between the variables.

## 4. Can cross terms have a negative impact on the interpretation of a line element?

Yes, cross terms can sometimes complicate the interpretation of a line element. They can introduce additional complexity and make it challenging to isolate the effects of individual variables on the system.

## 5. How can cross terms be controlled or manipulated in a line element?

Cross terms can be controlled or manipulated in a line element by using different mathematical techniques such as transformation or substitution. By manipulating the cross terms, it is possible to simplify the equation and make it easier to interpret.

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