A Student's Guide to Vectors and Tensors by Daniel Fleisch

By Daniel Fleisch

Vectors and tensors are one of the strongest problem-solving instruments on hand, with purposes starting from mechanics and electromagnetics to basic relativity. realizing the character and alertness of vectors and tensors is severely very important to scholars of physics and engineering. Adopting an analogous process utilized in his hugely renowned A Student's consultant to Maxwell's Equations, Fleisch explains vectors and tensors in undeniable language. Written for undergraduate and starting graduate scholars, the ebook offers a radical grounding in vectors and vector calculus sooner than transitioning via contra and covariant elements to tensors and their functions. Matrices and their algebra are reviewed at the book's aiding site, which additionally good points interactive strategies to each challenge within the textual content the place scholars can paintings via a chain of tricks or decide to see the complete resolution instantaneously. Audio podcasts supply scholars the chance to listen to very important recommendations within the publication defined through the writer.

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And as the angle between and and are parallel or approaches 90° or 270°, the magnitude of the cross product increases, reaching a maximum value of when the vectors are perpendicular. 9) Applying these relations term-by-term to the product of and should help you understand where Eqs. 5 come from (and if you need some help making that work out, there’s a problem on this at the end of this chapter, with the full solution on the book’s website). Applications of the cross product include torque problems (in which and magnetic force problems (in which you can find examples of these in the chapter-end problems.

1). 5 Partial derivatives Once you understand the basic vector operations of dot, cross, and triple products, it’s a small step to more advanced vector operations such as gradient, divergence, curl, and the Laplacian. But these are differential vector operations, so before you can make that step, it’s important for you to understand the difference between ordinary derivatives and partial derivatives. This is worth your time and effort because differential vector operations have many applications in diverse areas of physics and engineering.

15(b) is quite different. In this case, both subtracting from about this and are positive, and gives a small result. The value of the x-component of the curl is therefore small in this case. 42) A common misconception is that the curl of a vector field is non-zero wherever the field appears to curve. However, just as the divergence depended both on the spreading out and the changing length of field lines, the curl depends not only on the curvature of the lines but also on the strength of the field.

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