Cartesian to curvilinear coordinate transformations

In summary, the conversation discusses the use of vector notation in E&M problems and the difficulties of converting between different coordinate systems. It is suggested to use vector notation as much as possible and rewrite component formulas in terms of dot products or vector magnitudes to make it easier to select a suitable coordinate system for solving problems.
  • #1
Stendhal
24
1

Homework Statement


Is there a more intuitive way of thinking or calculating the transformation between coordinates of a field or any given vector?

The E&M book I'm using right now likes to use the vector field

## \vec F\ = \frac {\vec x} {r^3} ##

where r is the magnitude of ## \vec x ##In Cartesian coordinates, this looks like

## \frac {x \hat x + y \hat y + z \hat z} {\sqrt {x^2 + y^2 +z^2}^3} ##

In problems such as finding the flux through a sphere, it's difficult to use cartesian coordinates as it's very algrebra intensive, but I find it hard to convert between different coordinate systems. It also seems really unnecessary to simply look up the values of x,y,z and their respective ## \hat x ## directions for the components. Is there a better way to go about thinking and converting fields?
 
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  • #2
One of the advantages of using vector notation is that it should be independent of your coordinate system so to answer your question about "a more intuitive way..." I would say; Yes, use the vector notation as much as possible. Where you can, rewrite component formulas in terms of dot products or vector magnitudes. For example if you need the ##x##-component of a vector ##\vec{v}## write that as ##v_x=\hat{\imath} \bullet \vec{v}##. Granted you'll eventually need to resolve coordinates for such things as integrating over a surface but if you do most of your conceptual work in the general notation first, you often can select the coordinate system that makes this easiest.
 

What is a Cartesian coordinate system?

A Cartesian coordinate system is a mathematical system used to describe the position of a point in space. It is composed of three perpendicular axes (x, y, and z) that intersect at the origin (0, 0, 0). This system is commonly used in mathematics, physics, and engineering.

What is a curvilinear coordinate system?

A curvilinear coordinate system is a mathematical system used to describe the position of a point in space using curved coordinates instead of straight lines. This type of system is often used when describing objects or phenomena that have curved or irregular shapes, such as planets or fluid flow.

Why are Cartesian to curvilinear coordinate transformations important?

Cartesian to curvilinear coordinate transformations are important because they allow us to describe the same point in space using different coordinate systems. This can be useful when analyzing complex systems or solving mathematical problems that are easier to solve in one coordinate system compared to another.

What are some examples of Cartesian to curvilinear coordinate transformations?

Some examples of Cartesian to curvilinear coordinate transformations include polar coordinates, cylindrical coordinates, and spherical coordinates. These transformations involve converting the x, y, and z coordinates of a point to different coordinate systems based on their distance and angle from the origin.

How are Cartesian to curvilinear coordinate transformations used in science?

Cartesian to curvilinear coordinate transformations are used in various fields of science, such as physics, engineering, and geography. They are used to solve complex mathematical problems, analyze the behavior of physical systems, and create maps and models of the Earth's surface. They are also used in computer graphics and simulations to represent three-dimensional objects and environments.

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