ReviewofTutorial4-5 - NATIONAL UNIVERSITY OF SINGAPORE...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Revision Notes (Fourier Series and Vector Spaces) 5 Fourier Series Lecture Notes and Tutorial Question Review 1. The Formulas of Fourier Series. There are two kinds of formulas: one is for period 2 π ,theotherper iod2 L . The later one is much more general, and enough for all problems. f ( x )= a 0 + ± n =1 ( a n cos L x + b n sin L x ) , and the coefficients are given by the Euler’s formulas : a 0 = 1 2 L ² L L f ( x ) dx a n = 1 L ² L L f ( x )cos L xdx n =1 , 2 , ··· b n = 1 L ² L L f ( x )sin L n , 2 , The difficult part for fourier series is integration. 2. Trigonometric Functions. (a) The graph of cosine and sine functions. (b) Some special values of those function, for example sin 2 . (c) Some trigonometric identities. (d) Integration of these functions. (can use identities or integration by parts) 3. Parity. You need to know: (a) The advantage to consider parity Frst; (b) When f is an even function, which series will be applied? 4. Periodicity and Continuity. By deFnition, it is obvious that only periodic and piecewise continuous functions have their ±ourier series. However, for some kinds of partial functions, I mean a function is deFned on an interval, but not deFned on the other parts, then we can write out the ±ourier series of the function by using half-range expansions . Check details in Tutorial 4 Question 6. Also check tutorial notes 4 for more details about the advantage of half-range expansion compared with other expansions. 121
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
6 Vector Spaces 1. Arithmetic I skip the details here, since they are not difficult. You need to be very familiar with them. 2. Magnitude (Length) Given a vector u = u 1 i + u 2 j + u 3 k in 3-dimensional space, the length of the vector is given by ± u ± = ± u 2 1 + u 2 2 + u 2 3 . Remark : There are two kinds of notations for magnitude: ±± and || .(Th ef o rm e ri su s ed in your lecture notes; and the latter one is used in your textbook.) What’s the diFerence? No diFerence for 3-dimensional spaces. However, is more general, and it can be use in all cases to denote the length. is used only for some special spaces, for example, 3-dimensional spaces, 4-dimensional spaces, n -dimensional spaces. Application : Magnitude of a vector (like the module of a real number) does not have some good properties, for example, the function is not diFerentiable. Check Tutorial 5 Question 6. Some times we need to remove the magnitude by using the following property: ± u ± 2 = u · u .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 9

ReviewofTutorial4-5 - NATIONAL UNIVERSITY OF SINGAPORE...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online