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My differential geomtry professor once told me that with all his 30+ years experience in research in differential geometry he has never actually fully grasped what a tensor truly is.

I believe this is the case for many of us. You can memorize a definition, read many examples of the definition, and then even solve many problems successfully using the definition, and then think you fully understand the definition, but in fact you still may not fully understand what the definition really is. You may think you know what it is, but likely you understand what it really is less than you think.

Going back to my differential geometry professor, I try to give a description of what a tensor is without using mathematical terminology and I cannot, because I cannot feel what a tensor is either. Repeating the definition doesn't really help, and solving more problems won't help either, because I still cannot feel it. I find that solving problems using definitions helps you to improve your solving ability using the definitions, but only helps you to understand the definition a bit more, but not fully.

How do you know when you fully, fully understand a definition? Because solving problems correctly is not enough.

I believe this is the case for many of us. You can memorize a definition, read many examples of the definition, and then even solve many problems successfully using the definition, and then think you fully understand the definition, but in fact you still may not fully understand what the definition really is. You may think you know what it is, but likely you understand what it really is less than you think.

Going back to my differential geometry professor, I try to give a description of what a tensor is without using mathematical terminology and I cannot, because I cannot feel what a tensor is either. Repeating the definition doesn't really help, and solving more problems won't help either, because I still cannot feel it. I find that solving problems using definitions helps you to improve your solving ability using the definitions, but only helps you to understand the definition a bit more, but not fully.

How do you know when you fully, fully understand a definition? Because solving problems correctly is not enough.

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