GATE Chemical Engineering Engineering Mathematics Flashcards

24 sample cards from the Engineering Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Define an improper integral of the first kind and its convergence.

   An integral over an infinite interval, e.g. $\int_{a}^{\infty} f(x)\,dx = \lim_{t \to \infty} \int_{a}^{t} f(x)\,dx$. It converges if the limit exists and is finite; otherwise it diverges.

2. For what values of $p$ does $\int_{1}^{\infty} \frac{dx}{x^{p}}$ converge?

   It converges if $p > 1$ (to $\frac{1}{p-1}$) and diverges if $p \leq 1$.

3. State the property of a definite integral over symmetric limits for even and odd functions.

   $$\int_{-a}^{a} f(x)\,dx = \begin{cases} 2\int_{0}^{a} f(x)\,dx & f \text{ even} \\ 0 & f \text{ odd} \end{cases}$$

4. Define the Gamma function and give the value of $\Gamma(n)$ for a positive integer $n$.

   $$\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x}\,dx, \qquad \Gamma(n) = (n-1)! \text{ for integer } n \geq 1.$$ Also $\Gamma(1/2) = \sqrt{\pi}$.

5. What is a partial derivative of $f(x,y)$ with respect to $x$?

   It is the derivative treating $y$ as constant: $$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x,y)}{h}.$$

6. State the condition (Clairaut/Schwarz theorem) for equality of mixed partial derivatives.

   If the second partial derivatives are continuous, then the mixed partials are equal: $$\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}.$$

7. Write the total derivative (differential) of $z = f(x,y)$.

   $$dz = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy.$$

8. State the chain rule for the total derivative of $z = f(x,y)$ where $x$ and $y$ depend on $t$.

   $$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}.$$

9. State the first-order necessary condition for a local maximum or minimum of $f(x,y)$.

   At an interior extremum the first partial derivatives vanish: $$\frac{\partial f}{\partial x} = 0 \quad \text{and} \quad \frac{\partial f}{\partial y} = 0.$$ Such points are called critical (stationary) points.

10. State the second-derivative (Hessian) test for $f(x,y)$ at a critical point.

    Let $D = f_{xx}f_{yy} - f_{xy}^{2}$. If $D > 0$ and $f_{xx} > 0$: local minimum; $D > 0$ and $f_{xx} < 0$: local maximum; $D < 0$: saddle point; $D = 0$: test inconclusive.

11. What is the method of Lagrange multipliers used for?

    To find extrema of $f(x,y)$ subject to a constraint $g(x,y) = 0$ by solving $\nabla f = \lambda \nabla g$ together with $g = 0$, where $\lambda$ is the Lagrange multiplier.

12. Define the gradient of a scalar field $\phi(x,y,z)$.

    $$\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}.$$ It points in the direction of greatest rate of increase of $\phi$.

13. Define the divergence of a vector field $\vec{F} = F_1\hat{i} + F_2\hat{j} + F_3\hat{k}$.

    $$\nabla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}.$$ It is a scalar measuring net outward flux per unit volume.

14. Define the curl of a vector field $\vec{F}$.

    $$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \end{vmatrix}.$$ It is a vector measuring local rotation of the field.

15. State the vector identity for the divergence of a curl.

    $$\nabla \cdot (\nabla \times \vec{F}) = 0.$$ The divergence of any curl is identically zero.

16. State the vector identity for the curl of a gradient.

    $$\nabla \times (\nabla \phi) = \vec{0}.$$ The curl of any gradient is the zero vector (gradient fields are irrotational).

17. What is a solenoidal vector field and an irrotational vector field?

    A field is solenoidal if $\nabla \cdot \vec{F} = 0$ (divergence-free), and irrotational if $\nabla \times \vec{F} = \vec{0}$ (curl-free, i.e. conservative).

18. Give the formula for the directional derivative of $\phi$ in the direction of unit vector $\hat{a}$.

    $$D_{\hat{a}}\phi = \nabla \phi \cdot \hat{a}.$$ Its maximum value is $|\nabla \phi|$, attained along the direction of $\nabla \phi$.

19. What is the Laplacian of a scalar field $\phi$, and how is it related to gradient and divergence?

    $$\nabla^{2}\phi = \nabla \cdot (\nabla \phi) = \frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}} + \frac{\partial^{2}\phi}{\partial z^{2}}.$$

20. State Green's theorem in the plane.

    For a positively oriented simple closed curve $C$ bounding region $R$: $$\oint_{C} (P\,dx + Q\,dy) = \iint_{R}\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dx\,dy.$$

21. State Stokes' theorem.

    $$\oint_{C} \vec{F} \cdot d\vec{r} = \iint_{S} (\nabla \times \vec{F}) \cdot \hat{n}\,dS,$$ relating a line integral around closed curve $C$ to the surface integral of the curl over a surface $S$ bounded by $C$.

22. State the Gauss divergence theorem.

    $$\iint_{S} \vec{F} \cdot \hat{n}\,dS = \iiint_{V} (\nabla \cdot \vec{F})\,dV,$$ relating the flux of $\vec{F}$ through a closed surface $S$ to the volume integral of its divergence over $V$.

23. When is a first-order ODE $M(x,y)\,dx + N(x,y)\,dy = 0$ exact, and how is it solved?

    It is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. Then there exists $F$ with $F_x = M$, $F_y = N$, and the solution is $F(x,y) = c$.

24. Give the integrating factor and general solution form for the linear first-order ODE $\frac{dy}{dx} + P(x)y = Q(x)$.

    Integrating factor $\mu = e^{\int P\,dx}$. Solution: $$y \cdot e^{\int P\,dx} = \int Q\,e^{\int P\,dx}\,dx + c.$$

What this deck covers

Engineering Mathematics flashcards FAQ