GATE Mathematics Calculus Syllabus

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

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

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

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

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

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

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 $f$.

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 $f$.

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 the net outward flux per unit volume (source/sink strength).

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

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

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.

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.

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).

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

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

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

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

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.

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.

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 $S$.

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).

State Green's theorem in the plane.

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

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: $\oint_C \vec{F}\cdot \hat{n}\, ds = \iint_R (\nabla \cdot \vec{F})\, dA$.

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.

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.

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 $\hat{k}$.

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.

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

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

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

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

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: $\phi(A) - \phi(A) = 0$.

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.

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).

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 $\hat{u} = \cos\alpha\,\hat{i} + \sin\alpha\,\hat{j}$.

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

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

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

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

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.