![]() ![]() Two-Index Tensor Components as Products of Vector ComponentsĢ.9. Transformation Coefficients as Partial DerivativesĢ.5. Euclidean Vector Operations with and without Coordinatesġ.7. Derivatives of Euclidean Vectors with Respect to a Scalarġ.6. Euclidean Vectors, without Coordinatesġ.3. Why Aren't Tensors Defined by What They Are?ġ.2. It is an ideal companion for courses such as mathematical methods of physics, classical mechanics, electricity and magnetism, and relativity.ġ.1. Using a clear, step-by-step approach, the book strives to embed the logic of tensors in contexts that demonstrate why that logic is worth pursuing. Neuenschwander's Tensor Calculus for Physics is a bottom-up approach that emphasizes motivations before providing definitions. ![]() For example, in a classical mechanics course, they meet the "inertia tensor," and in electricity and magnetism, they encounter the "polarization tensor." However, this piecemeal approach can set students up for misconceptions when they have to learn about tensors in more advanced physics and mathematics studies (e.g., while enrolled in a graduate-level general relativity course or when studying non-Euclidean geometries in a higher mathematics class).ĭwight E. Undergraduate physics majors are typically introduced to tensors in special-case applications. This makes tensors the gold standard for expressing physical relationships in physics and geometry. Tensors, therefore, allow for the convenience of coordinates while also transcending them. But the true beauty of tensors lies in this fact: When coordinates are transformed from one system to another, tensors change according to the same rules as the coordinates. A horizontal electric field producing vertical polarization in dielectrics an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts-these are some situations where physicists employ tensors. Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. Now consider more complicated fields.Using a clear, step-by-step approach, this book explains one of the more difficult-yet crucial-topics in physics. One example of a vector field on a curved space is a weather map showing horizontal wind velocity at each point of the Earth's surface. Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. For example, the Riemann curvature tensor is a tensor field as it associates a tensor to each point of a Riemannian manifold, which is a topological space. Many mathematical structures called "tensors" are also tensor fields. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M. ![]() ![]() As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Not to be confused with the Tensor product of fields. ![]()
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