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ibkev
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I've seen the terms tensor calculus and tensor analysis both being used - what is the difference?
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History
The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century.[18]The word "tensor" itself was introduced in 1846 by William Rowan Hamilton[19] to describe something different from what is now meant by a tensor.[Note 3] The contemporary usage was introduced by Woldemar Voigt in 1898.[20]
Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892.[21] It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications).[22]
In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction ofEinstein's theory of general relativity, around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann.[23] Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect:
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Tensor Calculus is a branch of mathematics that deals with the study of tensors and their properties, such as differentiation and integration. Tensor Analysis, on the other hand, focuses on the geometric and algebraic properties of tensors, including their transformation rules and applications in physics and engineering.
Tensor Calculus is used in various fields such as physics, engineering, and computer science to model complex systems and analyze their behavior. Tensor Analysis is used in physics to study the properties of physical fields, such as gravitational and electromagnetic fields, and in engineering to analyze stress and strain in materials.
Yes, both Tensor Calculus and Tensor Analysis involve the study of tensors, which are mathematical objects that represent physical quantities with multiple components. They also both use similar notation and concepts, such as covariant and contravariant indices, to describe the properties of tensors.
An example of a tensor is the stress tensor, which is used in engineering to describe the distribution of forces within a material. In Tensor Calculus, the stress tensor can be used to calculate the strain and deformation of the material, while in Tensor Analysis it can be used to determine the principal stresses and directions of the material.
Yes, both Tensor Calculus and Tensor Analysis require a solid understanding of mathematics, particularly in areas such as linear algebra, multivariable calculus, and differential geometry. Without a strong foundation in these subjects, it can be challenging to grasp the concepts and applications of tensors in these fields.