GATE Mathematics Calculus Syllabus

Calculus syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Calculus in GATE Mathematics, not a summary of it.

1. 1

   Functions of two or more variables

   7 topics

   • Continuity
   • Directional derivatives
   • Partial derivatives
   • Total derivative
   • Maxima and minima
   • Saddle point
   • Method of Lagrange’s multipliers
2. 2

   Double and Triple integrals

   3 topics

   • Applications to area
   • Applications to volume
   • Applications to surface area
3. 3

   Vector Calculus

   8 topics

   • Gradient
   • Divergence
   • Curl
   • Line integrals
   • Surface integrals
   • Green’s theorem
   • Stokes’ theorem
   • Gauss divergence theorem

Calculus flashcards for GATE Mathematics

25 of 51 cards from the Calculus deck — real questions with worked answers.

1. State the $\varepsilon$-$\delta$ definition of continuity of a function $f$ at a point $a$.

   $f$ is continuous at $a$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|x - a| < \delta \implies |f(x) - f(a)| < \varepsilon$. Equivalently, $\lim_{x \to a} f(x) = f(a)$.

2. When is a function $f(x,y)$ said to be continuous at a point $(a,b)$?

   $f$ is continuous at $(a,b)$ if $\lim_{(x,y) \to (a,b)} f(x,y) = f(a,b)$, where the limit must exist and equal the function value along every path approaching $(a,b)$.

3. For functions of several variables, what is the relationship between continuity and existence of partial derivatives?

   Existence of all partial derivatives at a point does NOT imply continuity there. However, continuity does not imply differentiability either. Only differentiability (existence of the total derivative) implies continuity.

4. Define the partial derivative $\dfrac{\partial f}{\partial x}$ of $f(x,y)$ at $(a,b)$ using a limit.

   $$\frac{\partial f}{\partial x}(a,b) = \lim_{h \to 0} \frac{f(a+h, b) - f(a,b)}{h}$$ It is computed by differentiating with respect to $x$ while treating $y$ as a constant.

5. State Clairaut's (Schwarz's) theorem on mixed partial derivatives.

   If $f_{xy}$ and $f_{yx}$ are continuous on an open region, then they are equal: $$\frac{\partial^{2} f}{\partial x \, \partial y} = \frac{\partial^{2} f}{\partial y \, \partial x}$$

6. Define the directional derivative of $f$ at point $P$ in the direction of a unit vector $\hat{u}$.

   $$D_{\hat{u}} f(P) = \lim_{h \to 0} \frac{f(P + h\hat{u}) - f(P)}{h}$$ It measures the rate of change of $f$ in the direction $\hat{u}$.

7. How is the directional derivative computed from the gradient (for a differentiable $f$)?

   $$D_{\hat{u}} f = \nabla f \cdot \hat{u} = |\nabla f| \cos\theta$$ where $\hat{u}$ is a unit vector and $\theta$ is the angle between $\nabla f$ and $\hat{u}$.

8. In which direction is the directional derivative of $f$ maximum, and what is its maximum value?

   The directional derivative is maximum in the direction of $\nabla f$, and its maximum value equals $|\nabla f|$. It is minimum (most negative) in the direction $-\nabla f$, with value $-|\nabla f|$.

9. Define the total derivative (differentiability) of $f(x,y)$ at $(a,b)$.

   $f$ is differentiable at $(a,b)$ if $$f(a+h, b+k) - f(a,b) = f_x(a,b)\,h + f_y(a,b)\,k + \varepsilon_1 h + \varepsilon_2 k$$ where $\varepsilon_1, \varepsilon_2 \to 0$ as $(h,k) \to (0,0)$.

10. Write the total differential $df$ of a function $f(x,y,z)$.

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

11. State the chain rule for $\dfrac{dz}{dt}$ where $z = f(x,y)$ and $x = x(t),\ y = y(t)$.

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

12. What is the necessary (first-order) condition for a function $f(x,y)$ to have a local maximum or minimum at an interior point?

    All first-order partial derivatives must vanish: $f_x = 0$ and $f_y = 0$. Such a point is called a critical (stationary) point.

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

    If $D>0$ and $f_{xx}>0$: local minimum. If $D>0$ and $f_{xx}<0$: local maximum. If $D<0$: saddle point. If $D=0$: test inconclusive.

14. Define a saddle point of a function $f(x,y)$.

    A saddle point is a critical point where $f_x = f_y = 0$ but $f$ has neither a local maximum nor a local minimum; it increases along some directions and decreases along others. For the Hessian test, $D = f_{xx}f_{yy} - f_{xy}^{2} < 0$.

15. In the method of Lagrange multipliers, what condition is used to find extrema of $f(x,y,z)$ subject to constraint $g(x,y,z)=0$?

    $$\nabla f = \lambda \nabla g \quad \text{and} \quad g = 0$$ The gradients are parallel; $\lambda$ is the Lagrange multiplier.

16. Using Lagrange multipliers, what system is solved for extrema of $f$ subject to two constraints $g=0$ and $h=0$?

    $$\nabla f = \lambda \nabla g + \mu \nabla h, \quad g = 0, \quad h = 0$$ with multipliers $\lambda$ and $\mu$.

17. Geometrically, why does $\nabla f = \lambda \nabla g$ hold at a constrained extremum?

    At an extremum on the constraint surface $g=0$, the level surface of $f$ is tangent to the constraint surface, so their normal vectors $\nabla f$ and $\nabla g$ are parallel.

18. Write the double integral giving the area of a plane region $R$.

    $$\text{Area} = \iint_{R} dA = \iint_{R} dx\, dy$$

19. Write the formula for the area enclosed by a simple closed curve $C$ using a line integral (from Green's theorem).

    $$A = \frac{1}{2} \oint_{C} (x\, dy - y\, dx)$$

20. Write the triple integral giving the volume of a solid region $V$.

    $$\text{Volume} = \iiint_{V} dV = \iiint_{V} dx\, dy\, dz$$

21. Write the double integral for the volume of the solid under the surface $z = f(x,y)$ over a region $R$ (with $f \geq 0$).

    $$V = \iint_{R} f(x,y)\, dA$$

22. Write the formula for the surface area of $z = f(x,y)$ over a region $R$.

    $$S = \iint_{R} \sqrt{1 + \left(\frac{\partial f}{\partial x}\right)^{2} + \left(\frac{\partial f}{\partial y}\right)^{2}}\; dA$$

23. Write the formula for the surface area generated by revolving the curve $y=f(x)$, $a \le x \le b$, about the $x$-axis.

    $$S = \int_{a}^{b} 2\pi\, y \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}\; dx$$

24. Define the gradient operator $\nabla$ acting on a scalar field $f(x,y,z)$.

    $$\nabla f = \frac{\partial f}{\partial x}\,\hat{i} + \frac{\partial f}{\partial y}\,\hat{j} + \frac{\partial f}{\partial z}\,\hat{k}$$ It is a vector field pointing in the direction of steepest increase of $f$.

25. What is the geometric significance of the gradient $\nabla f$ relative to level surfaces?

    $\nabla f$ is normal (perpendicular) to the level surface $f = \text{constant}$ at each point, and points in the direction of greatest rate of increase of $f$.

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