Mathematical Tools for Physics
12—Tensors 393 12.8 What is the significance of a tensor satisfying the relationT ̃[T(~v)] =T[T ̃(~v)] =~vfor all~v? 12.9 Carry ...
12—Tensors 394 12.19 In three dimensions three non-collinear vectors from a point define a volume, that of the parallelepiped in ...
12—Tensors 395 12.26 Compute the transformation laws for the other components of the electromagnetic field tensor. 12.27 The div ...
Vector Calculus 2 There’s more to the subject of vector calculus than the material in chapter nine. There are a couple of types ...
13—Vector Calculus 2 397 Thendx=f ̇(t)dtanddy=g ̇(t)dt, so ds= √( f ̇(t)dt ) 2 + ( g ̇(t)dt ) 2 = √ f ̇(t)^2 +g ̇(t)^2 dt and th ...
13—Vector Calculus 2 398 Ifk→ 0 does this give the correct answer? Weighted Integrals The time for a particle to travel along a ...
13—Vector Calculus 2 399 Which one takes a shorter time? See problem 9. 3 What if the path is a parabola,x=y^2 .x 0 /y^20? It dr ...
13—Vector Calculus 2 400 here you need the whole vector from~rk− 1 to~rkin order to evaluate the work done as the mass moves fro ...
13—Vector Calculus 2 401 Example: IfF~=Axyxˆ+B(x^2 +L^2 )yˆ, what is the work done going from point(0,0)to(L,L)along the three d ...
13—Vector Calculus 2 402 Now divide this volume into a lot of little volumes,∆Vk with individual bounding surfacesSk. If you do ...
13—Vector Calculus 2 403 Use exactly the same reasoning that leads from the definition of the divergence to Eqs. ( 12 ) and ( 13 ...
13—Vector Calculus 2 404 Put all these pieces together and you have ˆn 1. ∮ S dA~×~v= ∮ C ~v.d~`∆h=ˆn 1 .curl~v∆A 1 ∆h Divide by ...
13—Vector Calculus 2 405 Multiply and divide each term in the sum ( 18 ) by∆Akand you have ∑ k [ 1 ∆Ak ∮ Ck ~v.d~` ] ∆Ak= ∮ C ~v ...
13—Vector Calculus 2 406 I need only theˆrcomponent of the curl because the surface integral uses only the normal (rˆ) component ...
13—Vector Calculus 2 407 The difference of these two integrals is ∫ 1 ~v.d~r− ∫ 2 ~v.d~r= ∮ ~v.d~r This equations happens becaus ...
13—Vector Calculus 2 408 is a function of the two endpoints alone. Fix~r 0 and treat this as a function of the upper limit~r. Ca ...
13—Vector Calculus 2 409 13.5 Reynolds’ Transport Theorem When an integral has limits that are functions of time, how do you dif ...
13—Vector Calculus 2 410 B~is a function of~rtoo, but I won’t write it. The first two terms have the same surface, so they combi ...
13—Vector Calculus 2 411 ∫ edge B~.dA~= ∫ C B~.d~`×~v∆t= ∫ C ~v×B~.d~`∆t (32) Put Eqs. ( 31 ) and ( 32 ) into Eq. ( 30 ) and the ...
13—Vector Calculus 2 412 Problems 13.1 In the equation ( 3 ) what happens if you start with a different parametrization forxandy ...
«
16
17
18
19
20
21
22
23
24
25
»
Free download pdf