Mathematical Tools for Physics - Department of Physics - University

1—Basic Stuff 14 however a unifying notation and language that lets you avoid writing down a lot of special cases. (Is it discre ...

1—Basic Stuff 15 The area of a circle is the sum of all the pieces of area within it ∫ dA= ∫R 0 rdr ∫ 2 π 0 dφ I find it more us ...

1—Basic Stuff 16 Is the function the sum or difference of two other much simpler functions? If so, you may find it easier to sk ...

1—Basic Stuff 17 OR, if you’re clever with partial fractions, you might realize that you can rearrangefas x a^2 −x^2 = − 1 / 2 x ...

1—Basic Stuff 18 20 Given that ∫∞ −∞dx/(1 +x (^2) ) =π,i.e.you don’t have to derive this, what then is∫∞ −∞dx/(α+x (^2) )? Now d ...

1—Basic Stuff 19 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 20 As a check, does this agree with the previous result forx=∞, Eq. (1.10)? 1.14 Use parametric differentiation to ...

1—Basic Stuff 21 are there? Because of its special importance later, look at the casee=f= 0and analyze it as if it’s a separate ...

1—Basic Stuff 22 1.33 From the definition of the Riemann integral make a numerical calculation of the integral ∫ 1 0 dx 4 1 +x^2 ...

1—Basic Stuff 23 1.44 Start from the definition of a derivative, manipulate some terms: (a) derive the rule for differen- tiatin ...

Infinite Series . Infinite series are among the most powerful and useful tools that you’ve encountered in your introductory calc ...

2—Infinite Series 25 Of course, even better than memorizing them is to understand their derivations so well that you can derive ...

2—Infinite Series 26 calculus books. Even better, when you understand the subject of complex variables, these questions about se ...

2—Infinite Series 27 The required observation is that an increasing sequence of real numbers, bounded above, has a limit. After ...

2—Infinite Series 28 Add these inequalities fromn=kton=∞and you get f(k) +f(k+ 1) +···> ∫k+1 k + ∫k+2 k+1 +···= ∫∞ k dxf(x) & ...

2—Infinite Series 29 Ask→∞this quotient approaches zero no matter the value ofx. This means that the series converges for allx. ...

2—Infinite Series 30 This is a geometric series, each of whose terms is itself an infinite series. It still beats plugging into ...

2—Infinite Series 31 Differentiate thismtimes with respect toxandntimes with respect toy, then setx=aandy=b. Only one term survi ...

2—Infinite Series 32 where it is a maximum. The largest contribution to the whole integral comes from the region near this point ...

2—Infinite Series 33 Asymptotic You may have noticed the symbol that I used in Eqs. (2.20) and (2.21). “∼” doesn’t mean “ap- pro ...