Explaining Tensors to Someone with Calculus Background

Tensors are indeed a fascinating and powerful concept in linear algebra, especially when applied to the field of differential geometry. While tensors can seem quite abstract at first, they can be readily understood by those with a solid foundation in calculus and linear algebra. This guide aims to break down the basics of tensors in a way that is accessible to those who have a decent understanding of calculus.

What are Tensors?

At their core, tensors are mathematical objects that extend the concepts of vectors and scalars. They are primarily a part of linear algebra, yet their applications span across various fields including physics, engineering, and computer science. Tensors can be thought of as multi-linear maps, which can act on one or more vectors to produce scalars or other tensors.

Basic Definitions

Rank-1 Contravariant Tensor: A vector is a rank-1 contravariant tensor. In simpler terms, a vector is an entity that has magnitude and direction and is used to represent physical quantities in space. Vectors in calculus and physics are typically denoted by boldface or with an arrow above the symbol, such as (vec{v}).

Rank-1 Covariant Tensor: A covector, also known as a 1-form, is a rank-1 covariant tensor. A covector can be thought of as a linear map from vectors to the real numbers. Mathematically, a covector (alpha) can be represented as a linear combination of basis covectors, such as (alpha a_1 alpha_1 a_2 alpha_2 ldots), where (a_i) are real numbers and (alpha_i) are the basis covectors.

Higher Ranks

Rank-2 Contravariant Tensor: A rank-2 contravariant tensor, often referred to as a bi-vector, is an entity formed by the outer product of two vectors. For example, (v wedge w) represents the bi-vector formed by (v) and (w). It can also be represented as a sum of such bi-vectors, such as ((v_1 w_1 v_2 w_2 ldots v_k w_k)).

Rank-2 Covariant Tensor: A rank-2 covariant tensor, or a bi-covector, can be thought of as a linear map from bi-vectors to the real numbers. It can also be viewed as a sum of pairs ((v, c)), where (v) is a vector and (c) is a covector.

General Rank Tensor: A tensor with (p) contravariant ranks and (q) covariant ranks is a linear map from (q)-vectors to (p)-vectors. It can also be represented as a sum of terms like ((v_1, v_2, ldots, v_p, c_1, c_2, ldots, c_q)), where each (v_i) is a vector and each (c_j) is a covector.

Calculus and Manifolds

Calculus comes into play when we start dealing with tensors on manifolds. A manifold is a topological space that locally resembles Euclidean space near each point. For a tensor to be defined on a manifold, it must be able to transform in a coordinated manner under a change of coordinates. This is where the concept of tangent vectors becomes crucial.

Tangent Vectors: A tangent vector at a point on a manifold represents the direction of a tangent line at that point. Tangent vectors are fundamental in differential geometry and are closely related to the concept of derivatives. For instance, if we consider a smooth function (f) on a manifold, the derivative of (f) at a point (p) can be interpreted as a tangent vector.

Tangent Manifold: The set of all tangent vectors at a point on a manifold forms a vector space called the tangent space. The collection of all tangent spaces at all points on the manifold forms the tangent manifold, denoted as (TM). The tangent manifold is itself a differentiable manifold, which means that it is smooth and continuous.

The complexity arises when we consider how tensors behave under changes of coordinates. This requires a deep understanding of concepts such as partial derivatives, differentiable manifolds, and the theory of differential forms. However, once these concepts are grasped, the analysis of tensors on manifolds becomes much more manageable.

Conclusion

Tensors are incredibly versatile mathematical objects that have numerous applications in various fields. While the initial understanding of tensors may require a solid foundation in linear algebra, their study can be greatly enhanced with a calculus background. By grasping the basic definitions and properties of tensors, one can appreciate their elegance and utility in more advanced mathematical and physical contexts.

From linear maps to manifolds, tensors provide a powerful framework for analyzing complex systems. By embracing the concepts discussed here, one can unlock a rich and rewarding exploration of tensor theory and its applications.