Contra-Variant Vector Transform: Taking Partials

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In summary, the contra-variant transform is defined by the differential transform from calculus, with dx^{\mu}=x^{\mu}_{,\nu}dx^{\nu} and A^{\mu}=x^{\mu}_{,\nu}A^{\nu}. These are not actual vectors or tensors, but rather functions that assign a tensor or vector to each point in space-time. Therefore, the partials can be taken at any point in space-time.
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exmarine
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The contra-variant transform seems to be defined by the differential transform from calculus.

dx[itex]^{\mu}[/itex]=x[itex]^{\mu}_{,\nu}[/itex]dx[itex]^{\nu}[/itex]

A[itex]^{\mu}[/itex]=x[itex]^{\mu}_{,\nu}[/itex]A[itex]^{\nu}[/itex]

I am puzzled by this, as the vector / tensor usually has finite components. They span a considerable region of space. So where are the partials to be taken, i.e., at what point in space or space-time?
 
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Those are NOT "vectors" or "tensors"- they are tensor or vector valued functions. That is they are functions that assign a tensor or vector to every point in space-time. Just as you do not take derivtives of numbers, but of functions, so the derivative is a function that can be evaluated at any point in space-time.
 

1. What is a Contra-Variant Vector Transform?

A Contra-Variant Vector Transform is a mathematical operation used in the field of vector calculus to transform a vector from one coordinate system to another. It involves taking the partial derivatives of the vector components with respect to the new coordinate system.

2. Why is a Contra-Variant Vector Transform important in scientific research?

The ability to transform vectors between different coordinate systems is crucial in many scientific fields, such as physics and engineering. It allows researchers to analyze and understand complex systems by breaking them down into simpler components.

3. How is a Contra-Variant Vector Transform different from a Co-Variant Vector Transform?

A Contra-Variant Vector Transform involves taking partial derivatives with respect to the new coordinate system, while a Co-Variant Vector Transform involves taking partial derivatives with respect to the original coordinate system. This results in different transformations and is applicable in different situations.

4. Can you provide an example of a Contra-Variant Vector Transform?

Sure, let's say we have a vector representing the velocity of an object in Cartesian coordinates (x, y, z). We want to transform this vector into spherical coordinates (r, θ, φ). Using the Contra-Variant Vector Transform, we would take the partial derivatives of the velocity components (Vx, Vy, Vz) with respect to the spherical coordinates (r, θ, φ) to obtain the transformed vector (Vr, Vθ, Vφ).

5. What are some practical applications of the Contra-Variant Vector Transform?

The Contra-Variant Vector Transform is used in various fields such as fluid mechanics, electromagnetism, and general relativity. It is used to transform velocity and force vectors, as well as to calculate the stress and strain tensors in materials. It is also used in computer graphics to transform 3D objects between different coordinate systems.

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