GATE Mathematics Calculus Flashcards

24 sample cards from the Calculus deck

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

1. What is $\nabla \cdot (\nabla \times \vec{F})$ for any vector field $\vec{F}$ with continuous second derivatives?

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

2. What is $\nabla \times (\nabla f)$ for any scalar field $f$ with continuous second derivatives?

   $$\nabla \times (\nabla f) = \vec{0}$$ The curl of a gradient is always zero; gradient fields are irrotational.

3. Define a solenoidal vector field and an irrotational vector field.

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

4. Write the Laplacian $\nabla^{2} f$ of a scalar field $f$ in Cartesian coordinates.

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

5. Write the general formula for the line integral of a vector field $\vec{F}$ along a curve $C$ parameterized by $\vec{r}(t)$, $a \le t \le b$.

   $$\int_{C} \vec{F} \cdot d\vec{r} = \int_{a}^{b} \vec{F}(\vec{r}(t)) \cdot \vec{r}\,'(t)\, dt$$

6. State the fundamental theorem for line integrals (gradient theorem).

   If $\vec{F} = \nabla \phi$, then $$\int_{C} \vec{F} \cdot d\vec{r} = \phi(B) - \phi(A)$$ where $A$ and $B$ are the endpoints of $C$. The integral is path-independent.

7. What is the condition for a line integral $\int_C \vec{F}\cdot d\vec{r}$ to be path-independent (conservative field)?

   $\vec{F}$ must be conservative, i.e. $\vec{F} = \nabla \phi$ for some scalar potential $\phi$. Equivalently (on a simply connected domain) $\nabla \times \vec{F} = \vec{0}$. Then $\oint_C \vec{F}\cdot d\vec{r} = 0$ for any closed curve.

8. Write the formula for the surface integral (flux) of a vector field $\vec{F}$ over a surface $S$.

   $$\iint_{S} \vec{F} \cdot \hat{n}\, dS = \iint_{S} \vec{F} \cdot d\vec{S}$$ where $\hat{n}$ is the unit outward normal to $S$.

9. For a surface $z = g(x,y)$ over region $R$, write the flux of $\vec{F}$ in terms of a double integral over $R$.

   $$\iint_{S} \vec{F} \cdot d\vec{S} = \iint_{R} \vec{F} \cdot \left(-g_x\,\hat{i} - g_y\,\hat{j} + \hat{k}\right) dx\, dy$$ (for upward orientation).

10. State Green's theorem in the plane.

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

11. State the circulation (curl) form and the flux (divergence) form of Green's theorem.

    Circulation form: $\oint_C \vec{F}\cdot d\vec{r} = \iint_R (\nabla \times \vec{F})\cdot \hat{k}\, dA$. Flux form: $\oint_C \vec{F}\cdot \hat{n}\, ds = \iint_R (\nabla \cdot \vec{F})\, dA$.

12. State Stokes' theorem.

    For an oriented surface $S$ with boundary curve $C$: $$\oint_{C} \vec{F} \cdot d\vec{r} = \iint_{S} (\nabla \times \vec{F}) \cdot \hat{n}\, dS$$ The line integral around the boundary equals the surface integral of the curl.

13. State the Gauss divergence theorem.

    For a closed surface $S$ enclosing a volume $V$: $$\oiint_{S} \vec{F} \cdot \hat{n}\, dS = \iiint_{V} (\nabla \cdot \vec{F})\, dV$$ The net outward flux equals the volume integral of the divergence.

14. How is Green's theorem related to Stokes' theorem?

    Green's theorem is the two-dimensional special case of Stokes' theorem, applied to a flat region in the $xy$-plane with the surface lying in that plane and normal $\hat{k}$.

15. Compare what each integral theorem relates: Green's, Stokes', and Gauss divergence.

    Green's: line integral around a plane curve $\leftrightarrow$ double integral over the enclosed plane region. Stokes': line integral around a space curve $\leftrightarrow$ surface integral of curl. Gauss: surface integral (flux) over a closed surface $\leftrightarrow$ volume integral of divergence.

16. Write the line integral of a scalar field $f$ with respect to arc length along curve $C$.

    $$\int_{C} f\, ds = \int_{a}^{b} f(\vec{r}(t))\, |\vec{r}\,'(t)|\, dt$$

17. Write the differential of arc length $ds$ for a plane curve $y = f(x)$.

    $$ds = \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}\; dx$$

18. For a conservative field $\vec{F} = \nabla \phi$, what is $\oint_C \vec{F}\cdot d\vec{r}$ around any closed curve $C$, and why?

    It equals $0$, because the line integral is path-independent and the start and end points coincide: $\phi(A) - \phi(A) = 0$.

19. State the second-order sufficient condition that distinguishes a saddle point from extrema for $f(x,y)$ in terms of the Hessian eigenvalues.

    At a critical point, if the Hessian matrix has both eigenvalues positive: local minimum; both negative: local maximum; eigenvalues of opposite sign (indefinite Hessian): saddle point.

20. What does a positive divergence $\nabla \cdot \vec{F} > 0$ at a point physically indicate?

    It indicates a source: there is net outward flux from that point (the field is expanding/diverging there). Negative divergence indicates a sink (net inward flux).

21. Express the directional derivative when $\hat{u}$ makes angle $\alpha$ with the $x$-axis in 2D, for $f(x,y)$.

    $$D_{\hat{u}} f = f_x \cos\alpha + f_y \sin\alpha$$ since $\hat{u} = \cos\alpha\,\hat{i} + \sin\alpha\,\hat{j}$.

22. Write the identity for the divergence of a product $\nabla \cdot (f \vec{F})$ where $f$ is scalar.

    $$\nabla \cdot (f\vec{F}) = f\,(\nabla \cdot \vec{F}) + \vec{F} \cdot \nabla f$$

23. Write the identity for the curl of a product $\nabla \times (f \vec{F})$ where $f$ is scalar.

    $$\nabla \times (f\vec{F}) = f\,(\nabla \times \vec{F}) + (\nabla f) \times \vec{F}$$

24. What is the relationship among differentiability, continuity, and the existence of the total derivative for multivariable functions?

    If $f$ is differentiable (the total derivative exists) at a point, then $f$ is continuous there and all partial derivatives exist. The converse fails: existence of partials does not guarantee differentiability or continuity.

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