By Stefan Hildebrandt, David Kinderlehrer, Mario Miranda
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Additional resources for Calculus of Variations and Partial Differential Equations. Proc. conf. Trento, 1986
2 ? 2 If you have access to a graphic calculator or a computer with suitable software, find out how to draw a curve from its polar equation. Check that you can adjust the scales so that in this case you get a circle, not just an ellipse. 2 Some graphic calculators will not draw the curve r = f(9) directly, but instead you can take 9 as a parameter and draw the curve with parametric equations x = f(0) cos 0, y = f(9) sin 9. Explain why this works. (i) Describe the motion of a point along the curve r = 1 + 2 cos 9 as 9 increases from 0 to 2yi.
Hence find C and S. eax sin bx dx (i) by using integration by parts twice (ii) by using the method of Question 10. Which method do you prefer? 48 Summations using complex numbers This section shows how complex numbers can be used to evaluate certain real sums. g. by induction, once you know the answer), but this is considerably more awkward. Sometimes it is worth setting out to do more than is required, as in the next example. (= n C n ) are right, but there are multiple angles, cos r9y instead of powers of cosines, cosr 9.
Hence show that a is a root of the cubic equation z3 + 7z2 + 15z + 25 = 0. (ii) Find the other two roots of this cubic equation. (iii) Illustrate the three roots of the cubic equation on an Argand diagram, and find the modulus and argument of each root. (iv) L is the locus of points in the Argand diagram representing complex numbers z for which Show that all three roots of the cubic equation lie on L and draw the locus L on your diagram. 6. 7. 8. 9. 9 36 EXERCISE 3B In Questions 1-6, draw an Argand diagram showing the set of points zfor which the given condition is true.
Calculus of Variations and Partial Differential Equations. Proc. conf. Trento, 1986 by Stefan Hildebrandt, David Kinderlehrer, Mario Miranda