Mathematics-II: Multiple Integrals, Differential Equations & Complex Analysis: 25 flashcards
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1. State Green's theorem in the plane.
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$\oint_C (M\,dx + N\,dy) = \iint_R (N_x - M_y)\,dA$, where $C$ is a closed curve enclosing region $R$.
2. What does it mean to 'verify' Green's or Stokes' theorem?
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Compute both sides of the theorem independently and show they are equal.
3. Stokes' theorem converts a line integral into what kind of integral?
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A surface integral of the curl of the vector field.
4. How is a triple integral evaluated?
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By nesting single-variable integrations in the correct order, treating other variables as constants.
5. What is the first step when changing the order of integration?
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Redraw the region of integration and rewrite the limits for the new order.
6. What form does Clairaut's equation take?
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$y = xp + f(p)$, where $p = dy/dx$.
7. How is Clairaut's equation solved?
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Differentiate with respect to $x$, solve for $p$, then substitute back to get general and singular solutions.
8. What substitution reduces $x^4 y' + x^3 y + \csc(xy) = 0$ to a separable form?
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$v = xy$.
9. For a constant-coefficient linear ODE, what two parts make up the general solution?
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Complementary function (from auxiliary equation) plus particular integral.
10. When is the method of variation of parameters used?
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When the right-hand side is non-standard (e.g. $\tan 2x$) and operator methods don't apply.
11. What substitution solves a Cauchy–Euler equation $x^2 y'' + axy' + by = f(x)$?
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$x = e^t$, which converts it to a constant-coefficient equation in $t$.
12. State the Cauchy–Riemann equations.
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$u_x = v_y$ and $u_y = -v_x$.
13. What is Milne–Thomson's method used for?
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Finding $v$ from $u$ (or $u$ from $v$) for an analytic function using the Cauchy–Riemann equations.
14. State Cauchy's integral formula.
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$\oint_C \frac{f(z)}{(z-a)^{n+1}}dz = \frac{2\pi i}{n!}f^{(n)}(a)$.
15. What does the residue theorem say?
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$\oint_C f(z)\,dz = 2\pi i \times$ (sum of residues at poles enclosed by $C$).
16. How is a real integral $\int_0^{2\pi} R(\cos\theta, \sin\theta)\,d\theta$ evaluated?
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Substitute $z = e^{i\theta}$ to convert it into a contour integral on the unit circle.
17. How is the residue at a pole of order $n$ computed?
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From the $(n-1)$-th derivative of $(z-a)^n f(z)$ evaluated at $a$, divided by $(n-1)!$.
18. What is the identity relating $|f(z)|^2$ and $|f'(z)|^2$?
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$(\partial_{xx} + \partial_{yy})|f(z)|^2 = 4|f'(z)|^2$.
19. In the Laurent expansion of $\frac{1}{(z-1)(z-2)}$, why do the three annuli give different series?
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Each annulus requires a different geometric series choice for $\frac{1}{z-1}$ and $\frac{1}{z-2}$ to ensure convergence.
20. What is the structure of each exam question in this paper?
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A compulsory 4-mark part (a) and a choice of two 8-mark parts from (b), (c), (d).
21. What is the main discriminator in this paper?
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Method selection — recognising which technique a given integral or equation demands.
22. What is an exact differential equation?
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One that can be solved once an integrating factor is found, making it exact.
23. What happens with repeated roots in the auxiliary equation?
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The complementary function includes $x e^{rx}$ terms.
24. Which theorem converts a line integral of a vector field to a surface integral of its curl?
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Stokes' theorem.
25. What does Cauchy's integral formula let you avoid?
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Direct integration — you evaluate derivatives at a point instead.
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