What are some practical applications of tensors in physics and mathematics?

[rjhollingsworth]

Hi all. This is my first post. w00t!

I'm just starting my sophmore year at university an I am trying to get ahead in physics, however I have hit a roadblock on tensors. I read all your posts in the 'what is a tensor' threads and I have a little better idea of what a tensor is, but I am still having trouble understanding them. The best way I come about understanding things I don't get in math/physics is by workin a problem containing the thing I do not understand. So do any of you have any problems where I would have to use a tensor? I am proficient in multivariable calc and have some experience in vector analysis (stokes theorem, divergence theorem, greens, etc) as well as matrix algebra (eigenvalues, eigenvectors, transformations, etc) to give you an idea where I am at as far as math goes. I would appreciate anything you can give me.

 

[robphy]

Try to start with the Moment of Inertia tensor:
http://maths.dur.ac.uk/~dma0cvj/mathphys/supplements/supplement2/supplement2.html
http://www.sfu.ca/~boal/211lecs/211lec23.pdf
http://baba.astro.cornell.edu/inertiatensor.pdf
http://theory.ph.man.ac.uk/~mikeb/lecture/pc167/rigidbody/contents.html

 

[M98Ranger]

Tensors have many practical applications in physics and mathematics. Here are a few examples:

1. General Relativity: Tensors are essential in the mathematical formulation of Einstein's theory of general relativity. They represent the curvature of spacetime and are used to describe the gravitational field and its effects on matter.

2. Fluid Dynamics: Tensors are used to describe the stress and strain in a fluid, which is important in understanding the behavior of fluids in motion. They are also used in the Navier-Stokes equations, which govern the motion of fluids.

3. Electromagnetism: Tensors are used to describe the electric and magnetic fields in Maxwell's equations. They also play a role in special relativity, where they are used to transform quantities between different frames of reference.

4. Quantum Mechanics: Tensors are used to represent physical quantities in quantum mechanics, such as spin and angular momentum. They are also used in quantum field theory to describe the interactions between particles.

5. Differential Geometry: Tensors are used to study the geometric properties of curved surfaces and manifolds. They are used in the study of differential equations and in the formulation of physical laws.

Overall, tensors are a powerful mathematical tool that allows us to describe and understand complex physical phenomena. They are used in a wide range of fields, from classical mechanics to quantum physics, and are essential for solving many problems in physics and mathematics. So, mastering tensors is crucial for any student pursuing a career in these fields.