Mathematical Tools for Physics
1—Basic Stuff 13 a b x 1 x 2 ξ 1 ξ 2 ξN PickN− 1 points betweenaandb. Call themx 1 ,x 2 , etc. a=x 0 < x 1 < x 2 <··· ...
1—Basic Stuff 14 1 2 x 1 /x To demonstrate this numerically, pick a function and do the first five steps explicitly. Pickf(x) = ...
1—Basic Stuff 15 The fundamental theorem of calculus unites the subjects of differentiation and integration. The integral is def ...
1—Basic Stuff 16 does it? It is. There is another definition that is worth knowing about, not because it helps you to do integra ...
1—Basic Stuff 17 To improve the sum, keep adding more and more points to the partition so that in the limit all the intervals xk ...
1—Basic Stuff 18 ∆m 8 =m(x 8 )−m(x 7 ) = (λx 8 +m 0 )−λx 7 =λL/10 +m 0. Choose the positionsx′kanywhere in the interval; for no ...
1—Basic Stuff 19 Where that last bracketed symbol means “greatest integer less than or equal tox”. It’s a notation more common i ...
1—Basic Stuff 20 1.7 Polar Coordinates When you compute an integral in the plane, you need the element of area appropriate to th ...
1—Basic Stuff 21 No surprise. b a For the preceding example you can do the double integral in either order with no special care. ...
1—Basic Stuff 22 How does the function behave as you approach the ends of the domain? If the domain extends from −∞to+∞, how do ...
1—Basic Stuff 23 −a a −a a Go back to the places that it blows up, and ask what happens near there. Ifxis a little greater than ...
1—Basic Stuff 24 Problems 1.1 What is the tangent of an angle in terms of its sine? Draw a triangle and do this in one line. 1.2 ...
1—Basic Stuff 25 1.11 What is the integral ∫∞ −∞dtt ne−t^2 ifn=− 1 orn=− 2? [Careful!, no conclusion-jumping allowed] 1.12 Sketc ...
1—Basic Stuff 26 1.19 Show that Γ(n+^1 / 2 ) = √ π 2 n (2n−1)!! The “double factorial” symbol mean the product of every other in ...
1—Basic Stuff 27 1.25 Derive all the limits on the integrals in Eq. ( 25 ) and then do the integrals. 1.26 Compute the area of a ...
1—Basic Stuff 28 1.32 Sketch by hand only, graphs of f(θ) = 1 + 1 2 sin^2 θ(0≤θ≤ 2 π), f(θ) = { θ ( 0 < θ < π) θ− 2 π (π & ...
1—Basic Stuff 29 1.40 If there are only 100 molecules of a gas bouncing around in a room, about how long will you have to wait t ...
Infinite Series Infinite series are among the most powerful and useful tools that you’ve been introduced to in an introductory c ...
2—Infinite Series 31 Some other common series that you need to know are power series for elementary functions: ex= 1 +x+ x^2 2! ...
2—Infinite Series 32 As you see from the last two examples you have to cast the problem into a form fitting the expansion that y ...
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