Are you ready to test my knowledge of tensors?

In summary, the conversation covers the topic of tensors, their definition and properties, and the differences between covariance and contravariance. The speaker mentions studying a book on tensors and finding it easy to understand, but also acknowledges the need to learn more about the theory behind tensors. They also discuss the modern treatment of tensors and how it has changed since the book was written in 1988. The conversation ends with a discussion about the importance of linear algebra in understanding tensors.
  • #36
But if the students did well in a basic physics class in high school, then they already got all of that, didn't they?

The point of making you take it again in the first year of college I thought was to show you how it's based in calculus? But using an integral, force differential x to find the work done is so simple, it reduces to an easy equation, F*x unless the force is a function of displacement. I spotted momentum as the derivative of kinetic energy with respect to velocity very easily. But really, I think someone who had not taken calculus but had good grounding in algebra and trigonometry would easily pass it. Is that a problem for most freshmen, remembering their algebra and trigonometry?

If I have a problem with something, it's usually because of an equation I don't know. Being too hardheaded to listen to how others approach the problem, I just try to figure out my own overcomplicated way. Is this how people are supposed to learn it? By having to figure out the equations all on their own? When it would be easier to just read a lengthy paper on how the equations were originally derived, a paper so lengthy you will never forget the equation that it is about? Should all children thus be kept away from advanced educational materials in case they spoil themselves for the mental exercise of knowledge deficiency? Come to think of it, that may be why it's so much easier for me than for other people, because I was knowledge deprived in the home for a good while. No internet, no books, I wrote down equations if I was lucky enough to see them on Science Channel. I borrowed a noncalculus book on physics and read it and played with equations in it just for fun. All of a sudden give me library books and internet, what happens?
 
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  • #37
CosmicKitten said:
But if the students did well in a basic physics class in high school, then they already got all of that, didn't they?
This is all irrelevant, let's get back to the topic of tensors.

What do you think of jgens questions above?
 
  • #38
Jorriss said:
This is all irrelevant, let's get back to the topic of tensors.

What do you think of jgens questions above?

The connection is Levi-Civita, torsion-free means a symmetric matrix which is why the two indices at the bottom of the Christoffel symbol commute, and proof of this concerns the development of the Christoffel symbols and why they are necessary, that much I know, but I'm too tired to think of a proof right now. I should be sleeping, and tomorrow I'm obligated to do stuff all day that does not allow me to study until... Thursday afternoon. I'll keep it on my mind until then, perhaps even come up with it sooner, although I'm a very different creature without my meds, as is the case on the days when I don't study...
 
  • #39
Closed pending moderation.
 
<h2>1. What are tensors?</h2><p>Tensors are mathematical objects that describe the relationship between different sets of data. They are often used in physics and engineering to represent physical quantities such as forces, velocities, and electric fields.</p><h2>2. How are tensors used in science?</h2><p>Tensors are used in a wide range of scientific fields, including physics, engineering, and computer science. They are particularly useful for describing physical phenomena that involve multiple dimensions, such as fluid dynamics, electromagnetism, and quantum mechanics.</p><h2>3. What is tensor calculus?</h2><p>Tensor calculus is a branch of mathematics that deals with the manipulation and analysis of tensors. It involves the use of calculus techniques to perform operations on tensors, such as differentiation and integration.</p><h2>4. How do tensors differ from vectors and matrices?</h2><p>Tensors are more general than vectors and matrices, as they can represent higher-dimensional data. Vectors are one-dimensional tensors, while matrices are two-dimensional tensors. Tensors can have any number of dimensions, making them more versatile for representing complex relationships between data sets.</p><h2>5. Are tensors difficult to understand?</h2><p>Tensors can be challenging to understand at first, as they involve abstract mathematical concepts and notation. However, with proper study and practice, they can be mastered like any other mathematical concept. Many online resources and textbooks are available to help with learning tensors.</p>

1. What are tensors?

Tensors are mathematical objects that describe the relationship between different sets of data. They are often used in physics and engineering to represent physical quantities such as forces, velocities, and electric fields.

2. How are tensors used in science?

Tensors are used in a wide range of scientific fields, including physics, engineering, and computer science. They are particularly useful for describing physical phenomena that involve multiple dimensions, such as fluid dynamics, electromagnetism, and quantum mechanics.

3. What is tensor calculus?

Tensor calculus is a branch of mathematics that deals with the manipulation and analysis of tensors. It involves the use of calculus techniques to perform operations on tensors, such as differentiation and integration.

4. How do tensors differ from vectors and matrices?

Tensors are more general than vectors and matrices, as they can represent higher-dimensional data. Vectors are one-dimensional tensors, while matrices are two-dimensional tensors. Tensors can have any number of dimensions, making them more versatile for representing complex relationships between data sets.

5. Are tensors difficult to understand?

Tensors can be challenging to understand at first, as they involve abstract mathematical concepts and notation. However, with proper study and practice, they can be mastered like any other mathematical concept. Many online resources and textbooks are available to help with learning tensors.

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