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Vance Grey
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https://www.physicsforums.com/attachments/205736
Not sure, whether this is correct what I'm going to say, because "calculus" as an English word is often used quite synonymous to "analysis" whereas in my language it means "system of calculation methods". So I interpret it in the sense of the latter and understand "tensor calculus" as computational methods where tensors are used. This could simply be how to deal with tensors, or a rather algebraic point of view if we consider tensor algebras and their categorical properties, or a topological point of view if we consider tensor fields, which are similar defined as vector fields, or analytically - and this is more likely - the connection of tensors on one hand, and tangent spaces, curvatures and differential forms on the other hand. Personally I would all of them summarize under "tensor calculus" and only spent a different amount of space for the different perspectives.Vance Grey said:https://www.physicsforums.com/attachments/205736
Link is broken...Vance Grey said:https://www.physicsforums.com/attachments/205736
Similar in English, actually, Calculus, sometimes 'Algebra' is used, as the rules used to operate with/on the objects.fresh_42 said:Not sure, whether this is correct what I'm going to say, because "calculus" as an English word is often used quite synonymous to "analysis" whereas in my language it means "system of calculation methods". So I interpret it in the sense of the latter and understand "tensor calculus" as computational methods where tensors are used. This could simply be how to deal with tensors, or a rather algebraic point of view if we consider tensor algebras and their categorical properties, or a topological point of view if we consider tensor fields, which are similar defined as vector fields, or analytically - and this is more likely - the connection of tensors on one hand, and tangent spaces, curvatures and differential forms on the other hand. Personally I would all of them summarize under "tensor calculus" and only spent a different amount of space for the different perspectives.
Tensor calculus is a branch of mathematics that deals with the study of tensors, which are multidimensional arrays used to describe the relationships between objects in a space. It is a powerful tool for understanding and analyzing complex systems in physics, engineering, and other scientific fields.
Tensors are mathematical objects that are used to represent physical quantities that have multiple components and are dependent on the coordinate system in which they are defined. They can be thought of as multidimensional arrays or matrices that are used to describe the relationships between different quantities in a given space.
Tensor calculus has a wide range of applications in various scientific fields, including physics, engineering, and computer science. It is used to model and analyze complex systems, such as fluid dynamics, electromagnetism, and general relativity. It also has applications in machine learning, image processing, and data analysis.
Tensor calculus can be a challenging topic to learn, as it requires a strong foundation in mathematics, including vector calculus and linear algebra. However, with dedication and practice, it can be mastered like any other mathematical concept.
Traditional calculus deals with functions of a single variable, whereas tensor calculus deals with functions of multiple variables. Additionally, tensor calculus involves the use of tensors and their operations, which are not commonly used in traditional calculus. It also has a more geometric interpretation, making it a useful tool for studying and understanding physical phenomena.