The number of components that a tensor will have in three dimensions depends on the rank of the tensor. In general, it will be N^R where N is the number of dimensions and R is the rank of the tensor.It is like a vector, but it has more components. Common tensors are the Inertia tensor, stress tensor, and strain tensor. Tensors have an algebra and a calculus. In three dimensions a tensor will have nine components. They have property called rank. Scalars are tensors of rank 0, vectors are tensors of rank 1, and the tensor of rank 2 is our inertia, stress, strain, or other quantity.
The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field, and the electromagnetic four-potential, (Φ,A), must be used instead.
Things like this, repeated time and time again, always amaze me and provide strong evidence, to me, that "math is real" and not just a human creation. We develop a set of mathematical models for a system that describes the system as we know it today, and then study the math behind the model and predict subtle behaviors that were previously completely unknown and, at times, seemingly impossible. Then, often decades later, those behaviors are demonstrated experimentally.
I read an article years ago titled "mathematical reality" that explored the subject so deeply that it even discussed its theological implications. Unfortunately, the link has been removed and I can't seem to find it anywhere.Things like this, repeated time and time again, always amaze me and provide strong evidence, to me, that "math is real" and not just a human creation. We develop a set of mathematical models for a system that describes the system as we know it today, and then study the math behind the model and predict subtle behaviors that were previously completely unknown and, at times, seemingly impossible. Then, often decades later, those behaviors are demonstrated experimentally.
It amazes me too and also makes my brain hurt trying to 'grok' it.Things like this, repeated time and time again, always amaze me and provide strong evidence, to me, that "math is real" and not just a human creation. We develop a set of mathematical models for a system that describes the system as we know it today, and then study the math behind the model and predict subtle behaviors that were previously completely unknown and, at times, seemingly impossible. Then, often decades later, those behaviors are demonstrated experimentally.
by Aaron Carman
by Aaron Carman
by Jake Hertz
by Jake Hertz