GATE 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 a basis of a vector space.

   A set of vectors that is both linearly independent and spans the entire space; its number of vectors equals the dimension of the space.

2. State the Rank–Nullity theorem for a linear map / matrix $A$ with $n$ columns.

   $\operatorname{rank}(A)+\operatorname{nullity}(A)=n$, where nullity is the dimension of the null space (solution space of $A\vec{x}=\vec{0}$).

3. Define the nullity of a matrix $A$.

   The nullity is the dimension of the null space (kernel) of $A$, i.e. the number of linearly independent solutions of $A\vec{x}=\vec{0}$, equal to the number of free variables.

4. State the (precise) definition of $\lim_{x\to a} f(x)=L$.

   For every $\varepsilon>0$ there exists $\delta>0$ such that $0<|x-a|<\delta$ implies $|f(x)-L|<\varepsilon$.

5. Give the three conditions for $f$ to be continuous at $x=a$.

   $f(a)$ is defined, $\lim_{x\to a}f(x)$ exists, and $\lim_{x\to a}f(x)=f(a)$.

6. What is the relationship between differentiability and continuity at a point?

   If $f$ is differentiable at $a$ then it is continuous at $a$; the converse is false (e.g. $f(x)=|x|$ at $x=0$ is continuous but not differentiable).

7. State the limit definition of the derivative $f'(x)$.

   $f'(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$.

8. State Rolle's theorem.

   If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c\in(a,b)$ with $f'(c)=0$.

9. State Lagrange's Mean Value Theorem.

   If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in(a,b)$ such that $f'(c)=\dfrac{f(b)-f(a)}{b-a}$.

10. State Cauchy's Mean Value Theorem.

    If $f,g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'\neq 0$ on $(a,b)$, then there exists $c\in(a,b)$ with $\dfrac{f'(c)}{g'(c)}=\dfrac{f(b)-f(a)}{g(b)-g(a)}$.

11. Write the Taylor series expansion of $f(x)$ about $x=a$.

    $f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^{n}=f(a)+f'(a)(x-a)+\dfrac{f''(a)}{2!}(x-a)^{2}+\cdots$

12. What is the Maclaurin series, and write the expansion of $e^{x}$.

    A Maclaurin series is a Taylor series about $a=0$: $e^{x}=\sum_{n=0}^{\infty}\dfrac{x^{n}}{n!}=1+x+\dfrac{x^{2}}{2!}+\dfrac{x^{3}}{3!}+\cdots$

13. State the first-derivative condition for a critical point and the second-derivative test for a local extremum of $f(x)$.

    Critical points satisfy $f'(x)=0$. At such a point: $f''(x)>0\Rightarrow$ local minimum, $f''(x)<0\Rightarrow$ local maximum, $f''(x)=0\Rightarrow$ test inconclusive.

14. For a function $f(x,y)$, state the necessary condition for a local extremum (stationary point).

    Both first partial derivatives vanish: $f_{x}=\dfrac{\partial f}{\partial x}=0$ and $f_{y}=\dfrac{\partial f}{\partial y}=0$.

15. State the second-derivative (Hessian) test for $f(x,y)$ using $D=f_{xx}f_{yy}-f_{xy}^{2}$.

    At a stationary point: if $D>0$ and $f_{xx}>0$ it is a local minimum; if $D>0$ and $f_{xx}<0$ a local maximum; if $D<0$ a saddle point; if $D=0$ the test is inconclusive.

16. Define the partial derivative $\dfrac{\partial f}{\partial x}$ of $f(x,y)$.

    $\dfrac{\partial f}{\partial x}=\lim_{h\to 0}\dfrac{f(x+h,y)-f(x,y)}{h}$ — differentiate with respect to $x$ treating $y$ as constant.

17. Define the gradient $\nabla f$ of a scalar field $f(x,y,z)$ and state its geometric meaning.

    $\nabla f=\left(\dfrac{\partial f}{\partial x},\dfrac{\partial f}{\partial y},\dfrac{\partial f}{\partial z}\right)$. It points in the direction of greatest increase of $f$ and is normal to level surfaces.

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

    $D_{\hat{u}}f=\nabla f\cdot\hat{u}$, where $\hat{u}$ is a unit vector. Its maximum value is $|\nabla f|$ (along $\nabla f$).

19. Define the divergence of a vector field $\vec{F}=(F_{1},F_{2},F_{3})$.

    $\operatorname{div}\vec{F}=\nabla\cdot\vec{F}=\dfrac{\partial F_{1}}{\partial x}+\dfrac{\partial F_{2}}{\partial y}+\dfrac{\partial F_{3}}{\partial z}$ — a scalar measuring net outflow per unit volume.

20. Define the curl of a vector field $\vec{F}$ and what it measures.

    $\operatorname{curl}\vec{F}=\nabla\times\vec{F}=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\ \partial_x&\partial_y&\partial_z\\ F_1&F_2&F_3\end{vmatrix}$; it measures the local rotation (circulation density) of the field.

21. What are the identities $\operatorname{curl}(\nabla f)$ and $\operatorname{div}(\operatorname{curl}\vec{F})$ equal to?

    $\nabla\times(\nabla f)=\vec{0}$ and $\nabla\cdot(\nabla\times\vec{F})=0$ (curl of a gradient and divergence of a curl are always zero).

22. State the Fundamental Theorem of Calculus (evaluation form).

    If $F'(x)=f(x)$ on $[a,b]$, then $\displaystyle\int_{a}^{b}f(x)\,dx=F(b)-F(a)$.

23. Write the integration by parts formula.

    $\displaystyle\int u\,dv=uv-\int v\,du$.

24. What is an improper integral, and what does it mean for it to converge? Illustrate with $\int_{1}^{\infty}\frac{1}{x^{p}}\,dx$.

    An improper integral has an infinite limit or unbounded integrand; it converges if its limit exists (is finite). $\displaystyle\int_{1}^{\infty}\frac{dx}{x^{p}}$ converges if $p>1$ and diverges if $p\leq 1$.

What this deck covers

Engineering Mathematics flashcards FAQ