By Daniel Fleisch
Vectors and tensors are one of the strongest problem-solving instruments on hand, with purposes starting from mechanics and electromagnetics to normal relativity. realizing the character and alertness of vectors and tensors is seriously very important to scholars of physics and engineering. Adopting a similar method utilized in his hugely well known A Student's consultant to Maxwell's Equations, Fleisch explains vectors and tensors in simple 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 parts to tensors and their functions. Matrices and their algebra are reviewed at the book's assisting web site, which additionally good points interactive options to each challenge within the textual content the place scholars can paintings via a chain of tricks or decide to see the full answer right now. Audio podcasts provide scholars the chance to listen to very important suggestions within the e-book defined by means of the writer.
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Additional resources for A Student's Guide to Vectors and Tensors
That means that since the negative of a vector is just a vector of the same magnitude in the opposite direction. 2 Direction of the cross product × . 3 The cross product as area. 3. Just as the dot product involves the projection of one vector onto another, the cross product also has a geometrical interpretation. In this case, the magnitude of the cross product between two vectors is proportional to the area of the parallelogram formed with those two vectors as adjacent sides. As you may recall, the area of a parallelogram is just its base times its height, and in this case the height of the parallelogram is sin(θ) and the length of the base is .
19 Function Á (varying as 1/r) and the gradient and contours of Á near the peak. 20 Function Á (varying as –1/r) and the gradient and contours of Á near the bottom of the valley. 20(b). Near the bottom of a valley, the gradient “flows” outward in all directions, so the divergence of the gradient is a large positive number in this case (again consistent with the conclusion that the Laplacian of a minimum point is positive). And what is the value of the Laplacian of a function away from a peak or valley?
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. 3 Triple scalar product Once you understand the dot product and cross product described in the previous two sections, you may be wondering if it’s possible to combine these two vector operations.
A Student's Guide to Vectors and Tensors by Daniel Fleisch