Conversion between vector components in different coordinate systems

In summary, the conversation discusses the formulas for translating between different coordinate systems. The equations ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## are used to translate between Cartesian and non-Cartesian coordinate systems. The second equation, ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}##, shows the relationship between the b:th and j:th components of a vector expressed in tangent vectors. However, there is some disagreement about the correct form of the equations, with the suggestion that they should
  • #1
Karl Karlsson
104
12
Homework Statement
I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean? Is this just the relation between the b:th and j:th component of ##\vec v## being expressed as tangent vectors?
Relevant Equations
##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}##
I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean? Is this just the relation between the b:th and j:th component of ##\vec v## being expressed as tangent vectors?
 
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  • #2
Hmm. The "relevant equations" seem slightly wrong, to me. I think it should be:

##v^j = v^a \frac{\partial x^j}{\partial \chi^a}## and ##v^a = v^j \frac{\partial \chi^a}{\partial x^j}##

Having said that, the equations are good for translating between any two coordinate systems, whether Cartesian, or not.
 
  • #3
I'm surprised that this question is considered "precalculus". It seems like calculus, to me.
 
  • #4
stevendaryl said:
I'm surprised that this question is considered "precalculus". It seems like calculus, to me.
To me, also. I have moved this thread, although the OP might not still be interested in it.
 

1. What are vector components?

Vector components are the individual parts of a vector that describe its magnitude and direction. They are typically represented by the x, y, and z axes in a three-dimensional coordinate system.

2. Why is it important to convert between vector components in different coordinate systems?

Converting between vector components allows us to describe the same vector in different coordinate systems, making it easier to compare and analyze vectors from different perspectives. It also allows us to apply mathematical operations to vectors in different coordinate systems.

3. How do you convert between vector components in different coordinate systems?

To convert between vector components in different coordinate systems, you can use trigonometric functions and basic vector operations such as addition, subtraction, and scalar multiplication. It is important to understand the relationships between the different coordinate systems and how to use them in calculations.

4. What are the most common coordinate systems used for vector components?

The most common coordinate systems used for vector components are Cartesian coordinates, polar coordinates, and spherical coordinates. Cartesian coordinates use x, y, and z axes, polar coordinates use a radius and angle, and spherical coordinates use a radius, inclination angle, and azimuth angle.

5. Are there any tools or software that can help with converting between vector components in different coordinate systems?

Yes, there are many tools and software available that can help with converting between vector components in different coordinate systems. Some examples include MATLAB, Wolfram Alpha, and various online calculators. It is important to understand the concepts and calculations involved in conversion, but these tools can make the process quicker and more accurate.

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