Kit Library / Mathematics / Engineering Mathematics
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Swap dx and dy to crack hard integrals.
When a double integral is hard to evaluate in the given order, changing the order of integration often simplifies it by reversing the limits and integrand order.
↳ Changing order of integration is a powerful strategy when the original order leads to impossible antiderivatives.
Line integral equals double integral.
↳ Green's theorem converts a line integral into a double integral (or vice versa) for simply connected regions.
Surface integral meets line integral.
↳ Stokes' theorem generalizes Green's theorem to 3D surfaces and is used to verify circulation-curl relationships.
Make any linear ODE exact.
For a first-order linear ODE of the form $\frac{dy}{dx} + P(x)y = Q(x)$, multiply by the integrating factor $\mu(x) = e^{\int P(x)\,dx}$ to make it exact.
↳ The integrating factor method transforms a non-exact linear ODE into an exact one that can be integrated directly.
Evaluate complex integrals instantly.
↳ Cauchy's integral formula lets you compute derivatives of analytic functions using contour integrals.
Missing Jacobian? Your integral is wrong.
↳ The Jacobian accounts for area scaling when transforming coordinates; omitting it gives incorrect results.
Check both sides to confirm the theorem.
↳ Verification requires evaluating both the line integral and the double integral and showing they are equal.
Solve non-homogeneous ODEs with ease.
For a second-order linear ODE $y'' + P(x)y' + Q(x)y = R(x)$, first find the complementary solution $y_c = c_1 y_1 + c_2 y_2$, then use variation of parameters to find a particular solution.
↳ Variation of parameters works for any non-homogeneous term, unlike undetermined coefficients which is limited to specific forms.
Complex differentiability is stricter.
↳ Analyticity requires differentiability in an open set, not just at a point, and implies the Cauchy-Riemann equations hold.
Series with negative powers.
↳ Laurent series generalize Taylor series to include negative powers, essential for analyzing singularities.
Secure easy marks upfront.
In this exam pattern, part (a) of each question is compulsory and carries 4 marks. Always attempt these first to secure easy marks before moving to the longer 8-mark parts.
↳ Prioritize compulsory short-answer parts to build momentum and secure marks.
IF = e^∫P dx.
For $\frac{dy}{dx} + P(x)y = Q(x)$, the integrating factor is always $e^{\int P\,dx}$. Just remember: IF = e to the integral of P.
↳ The integrating factor formula is a quick recall that saves time in exams.
Is the point inside the contour?
↳ Always check the location of singularities relative to the contour before applying Cauchy's formula.
From fluid flow to signal processing.
Complex analysis is used in engineering to model fluid flow, electromagnetic fields, and signal processing. Analytic functions represent potential flows, and contour integration solves real integrals.
↳ Complex analysis provides powerful tools for solving real-world engineering problems.
Sum residues for quick integrals.
For integrals over closed contours, the residue theorem states $\oint_C f(z)\,dz = 2\pi i \sum \text{Res}(f, z_k)$ where $z_k$ are singularities inside $C$. This is often faster than direct integration.
↳ The residue theorem simplifies many contour integrals to a sum of residues.
This topic covers key techniques from a B.Tech Mathematics-II syllabus: evaluating multiple integrals and changing order of integration, verifying Green's and Stokes' theorems, solving first-order and higher-order differ…
Full explanation, worked examples & memory tricks in the app
All 21 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. Which theorem converts a line integral around a closed curve into a double integral over the enclosed region?
BeginnerD · Green's theorem
Green's theorem states $\oint_C (M\,dx + N\,dy) = \iint_R (N_x - M_y)\,dA$, converting a line integral to a double integral.
Q2. In the exam pattern described, what is the mark distribution for each question?
BeginnerD · Compulsory 4-mark part (a) and choice of two 8-mark parts
Each question has a compulsory 4-mark part (a) and a choice of two 8-mark parts from (b), (c), (d).
Q3. What is the first step when changing the order of integration in a multiple integral?
BeginnerD · Redraw the region and rewrite the limits
Changing the order requires redrawing the region of integration and rewriting the limits for the new order.
Q4. Which substitution reduces $x^4 y' + x^3 y + \csc(xy) = 0$ to a separable form?
BeginnerB · $v = xy$
The substitution $v = xy$ collapses the equation into a separable form.
Q5. For a constant-coefficient linear ODE, the general solution is the sum of which two parts?
BeginnerD · Complementary function and particular integral
The general solution is the complementary function (from the auxiliary equation) plus a particular integral.
Q6. Which substitution converts a Cauchy–Euler equation into a constant-coefficient equation?
BeginnerB · $x = e^t$
Substituting $x = e^t$ converts $x^2 y'' + axy' + by = f(x)$ into a constant-coefficient equation in $t$.
Q7. What are the Cauchy–Riemann equations for $f(z) = u + iv$?
BeginnerD · $u_x = v_y$ and $u_y = -v_x$
A function is analytic iff the Cauchy–Riemann equations $u_x = v_y$ and $u_y = -v_x$ hold.
Q8. Cauchy's integral formula evaluates $\oint_C \frac{f(z)}{(z-a)^{n+1}}dz$ as which expression?
BeginnerD · $\frac{2\pi i}{n!}f^{(n)}(a)$
Cauchy's integral formula gives $\oint_C \frac{f(z)}{(z-a)^{n+1}}dz = \frac{2\pi i}{n!}f^{(n)}(a)$.
Q9. What does the residue theorem state about a contour integral?
BeginnerA · It equals $2\pi i$ times the sum of enclosed residues
The residue theorem states $\oint_C f(z)\,dz = 2\pi i \times$ (sum of residues at enclosed poles).
Q10. To evaluate $\int_0^{2\pi} R(\cos\theta, \sin\theta)\,d\theta$, which substitution is used?
BeginnerB · $z = e^{i\theta}$
The substitution $z = e^{i\theta}$ converts the real integral into a contour integral on the unit circle.
Q11. Cauchy's integral formula $\oint_C \frac{f(z)}{(z-a)^{n+1}}dz = \frac{2\pi i}{n!}f^{(n)}(a)$ lets you evaluate certain contour integrals without any integration.
BeginnerA · True
Cauchy's integral formula $\oint_C \frac{f(z)}{(z-a)^{n+1}}dz = \frac{2\pi i}{n!}f^{(n)}(a)$ lets you evaluate certain contour integrals without any integration.
Q12. A student needs to solve $(D^2+4)y = \tan 2x$. Which method is most appropriate?
IntermediateA · Variation of parameters
The right-hand side $\tan 2x$ is non-standard, so variation of parameters is used.
Q13. In the Laurent expansion of $\frac{1}{(z-1)(z-2)}$, why does the series differ across $|z|<1$, $1<|z|<2$, and $|z|>2$?
IntermediateC · Each annulus requires a different geometric series for convergence
Each annulus demands a different geometric series choice for $\frac{1}{z-1}$ and $\frac{1}{z-2}$ to ensure convergence.
Q14. Given $u = \log\sqrt{x^2+y^2}$, which method finds $v$ such that $f = u+iv$ is analytic?
IntermediateB · Milne–Thomson's method
Milne–Thomson's method uses the Cauchy–Riemann equations to find $v$ from $u$.
Q15. What does 'verify' mean when applied to Green's or Stokes' theorem in an exam?
IntermediateB · Compute both sides independently and show equality
Verification means computing both sides of the theorem separately and demonstrating they are equal.
Q16. For the equation $y = xp + f(p)$, what is the first step in finding the solution?
IntermediateB · Differentiate with respect to $x$
Clairaut's equation is solved by differentiating with respect to $x$, solving for $p$, then substituting back.
Q17. Evaluate the triple integral $$\int_0^1 \int_0^1 \int_0^{1-x-y} (x+y+z) \, dz \, dy \, dx$$ by changing the order of integration. What is the final result?
AdvancedC · 1/3
Changing the order of integration requires redrawing the region. The correct limits yield the result of 1/3 after evaluating the integral.
Q18. Given the differential equation $y' + 2y = e^{-x}$, find the particular solution using the method of integrating factors. What is the particular solution?
AdvancedC · y = e^{-2x}(C + e^{-x})
The integrating factor is $e^{2x}$. Multiplying through gives the left side as a derivative, allowing us to solve for $y$ and find the particular solution.
Q19. For the function $f(z) = z^2 + 2z + 1$, find the residue at the pole $z = -1$ and evaluate the integral $$\oint_C f(z) \, dz$$ where $C$ is a contour enclosing the pole.
AdvancedC · 1
The residue at $z = -1$ is found by evaluating the limit of $(z + 1)f(z)$ as $z$ approaches -1, which equals 1. By the residue theorem, the integral evaluates to $2\pi i$ times this residue.
Q20. Using the Cauchy-Riemann equations, determine if the function $f(z) = e^{x}(\cos y + i\sin y)$ is analytic. What can be concluded?
AdvancedB · It is not analytic anywhere.
The Cauchy-Riemann equations do not hold for this function, indicating it is not analytic anywhere in the complex plane.
Q21. Solve the Cauchy–Euler equation $$x^2 y'' + 3xy' + 4y = 0$$. What is the general solution?
AdvancedC · y = C_1 x^{-1} + C_2 x^{-4}
The substitution $x = e^t$ transforms the equation into a constant-coefficient form, leading to the characteristic equation with roots that yield the general solution.
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