Engineering Mathematics Short Notes | Vector Calculus: Gradient, Divergence, Curl, Line Integrals & Green’s Theorem

VECTOR CALCULUS – COMPLETE NOTES (ENGINEERING MATHEMATICS) for GATE and University Exams

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1. Introduction to Vector Calculus

Vector calculus deals with differentiation and integration of vector-valued functions. It is extensively used in:

• Fluid mechanics

• Electromagnetic field theory

• Heat transfer

• Robotics and control

• Computer graphics

• Optimization problems

The fundamental operators in vector calculus are:

• Gradient (∇f)

• Divergence (∇·F)

• Curl (∇×F)

• Laplacian (∇²f) (used in PDEs)

The base operator is the del operator:

$$\mathrm{\nabla }=\widehat{i}\frac{\mathrm{\partial }}{\mathrm{\partial }x}+\widehat{j}\frac{\mathrm{\partial }}{\mathrm{\partial }y}+\widehat{k}\frac{\mathrm{\partial }}{\mathrm{\partial }z}$$

This operator acts differently depending on whether it is applied to a scalar or a vector field.

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2. Gradient (grad)

Definition

For a scalar function 

$f(x, y, z)$ the gradient is:

$$\mathrm{\nabla }f=\widehat{i}\frac{\mathrm{\partial }f}{\mathrm{\partial }x}+\widehat{j}\frac{\mathrm{\partial }f}{\mathrm{\partial }y}+\widehat{k}\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathrm{z}}$$

The gradient gives a vector field.

Interpretation

• Points in the direction of maximum increase of the function.

• Magnitude gives maximum rate of change.

Key Properties

1. Directional derivative in direction of unit 

2. vector $\hat{u}$:

$$D_{\widehat{u}}f=\mathrm{\nabla }f\cdot \widehat{u}$$

2. Gradient is perpendicular to level surfaces
   For scalar field 

   $f(x,y,z) = c$, the normal direction is $\nabla f$.

3. If gradient is zero, function is constant.

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Example

Let

$$f = x^2 + y^2 + z^2$$
$$\nabla f = (2x)\hat{i} + (2y)\hat{j} + (2z)\hat{k}$$

Gradient points radially outward from origin.

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3. Divergence (div)

Definition

For a vector field

$$\mathbf{F}=F_x\widehat{i}+F_y\widehat{j}+F_z\widehat{k}$$

the divergence is:

$$\nabla\cdot\mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$gIt gives a scalar.

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Physical Meaning

Divergence measures net outward flux of a vector field (like fluid flow).

• Positive divergence → source region

• Negative divergence → sink region

• Zero divergence → incompressible flow / solenoidal field

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Example

$$\mathbf{F} = x^2\hat{i} + y^2\hat{j} + z^2\hat{k}$$
$$\mathrm{\nabla }\cdot \mathbf{F}=2x+2y+2z$$

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4. Curl (rot)

Definition

For a vector field 

$\mathbf{F}$:

$$\mathrm{\nabla }\times \mathbf{F}=\mid \begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ \frac{\mathrm{\partial }}{\mathrm{\partial }x}& \frac{\mathrm{\partial }}{\mathrm{\partial }y}& \frac{\mathrm{\partial }}{\mathrm{\partial }z}\\ F_x& F_y& F_z\end{array}\mid <span\; class="mopen"><span\; class="delimsizing\; mult"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist-s"></span></span></span></span></span><span\; class="mclose"><span\; class="delimsizing\; mult"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist-s"></span></span></span></span></span>$$

It gives a vector.

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Physical Meaning

Curl measures the rotational tendency of a vector field.

• Zero curl → irrotational field

• If $\mathrm{\nabla }\times F=\mathrm{0,\; the\; field\; is\; gradient\; of\; some\; scalar\; potential.}$

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Example

$$\mathbf{F}=yz\hat{i}+zx\hat{j}+xy\hat{k}$$
$$\mathrm{\nabla }\times \mathbf{F}=(x-x)\widehat{i}+(y-y)\widehat{j}+(z-z)\widehat{k}=0$$

So field is irrotational.

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5. Important Vector Identities (Very Important for GATE)

$$\nabla\cdot(\nabla\times\mathbf{F}) = 0$$
$$\nabla\times(\nabla f) = 0$$
$$\nabla\cdot(f\mathbf{F}) = f(\nabla\cdot\mathbf{F}) + \mathbf{F}\cdot(\nabla f)$$
$$\nabla\times(f\mathbf{F}) = f(\nabla\times \mathbf{F}) + (\nabla f)\times\mathbf{F}$$
$$\mathrm{\nabla }(fg)=f\mathrm{\nabla }g+g\mathrm{\nabla }f$$

Laplacian

$${\mathrm{\nabla }}^{2}f=\mathrm{\nabla }\cdot (\mathrm{\nabla }f)$$

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6. Line Integrals

Let $\mathbf{F} = P\hat{i} + Q\hat{j} + R\hat{k}$
Line integral along curve $C$:

$${\int }_C\mathbf{F}\cdot d\mathbf{r}={\int }_{C}Pdx+Qdy+Rdz$$

Interpretation

• Work done by force field

• Circulation of a vector field

• Used in electromagnetics

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Properties

1. Path dependent unless field is 

2. conservative.

3. If $\nabla \times F = 0$, then:

$$\int_C F\cdot dr = \phi(B)-\phi(A)$$


3. For closed curve (C is closed):

$${\oint }_{C}F\cdot dr$$

represents total circulation.

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Example

Find

$${\int }_C(xdx+ydy)$$

from $(0,0)$ to $(1,1)$ along straight line.

Parametrize:

$$x=t,\,y=t,\, 0\le t\le 1$$
$$dx=dt,\,dy=dt$$
$${\int }_0^{1}tdt+tdt=2{\int }_0^{1}tdt=1$$

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7. Green’s Theorem in Plane

Statement

For continuously differentiable P and Q:

$${\oint }_C(Pdx+Qdy)={\iint }_R(\frac{\mathrm{\partial }Q}{\mathrm{\partial }y}-\frac{\mathrm{\partial }P}{\mathrm{\partial }x})dA$$

Where:

• C = positively oriented (counter-clockwise) boundary curve

• R = region enclosed by C

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Interpretation

Converts:

• A line integral → area (double) integral

This is extremely important in GATE (2–4 marks frequently).

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Useful Form

$$\oint_C x\,dy = \iint_R 1\,dA = \text{Area(R)}$$
$$\oint_C -y\,dx = \text{Area(R)}$$

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Example

Evaluate

$${\oint }_C(x^2-y^2)dx+(xy)dy$$

Using Green’s Theorem:

$$\iint_R \left(\frac{\partial}{\partial y}(xy)-\frac{\partial}{\partial x}(x^2-y^2)\right)\,dA$$$$={\iint }_R(x-2x)dA={\iint }_R(-x)dA$$

For a symmetric region around y-axis → integral = 0.

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8. More Examples

Example – Gradient

$$f=xy{e}^z$$$$\mathrm{\nabla }f=(y{e}^z)\widehat{i}+(x{e}^z)\widehat{j}+(xy{e}^z)\widehat{k}$$

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Example – Divergence

$$F = (3x^2 + y)\hat{i} + (x + 4y)\hat{j}$$ $$\nabla \cdot F = 6x + 4$$

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Example – Curl

$$F = (y, z, x)$$
$$\nabla\times F = (1-0)\hat{i} + (0-1)\hat{j} + (1-1)\hat{k} = (1, -1, 0)$$

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Example – Green’s Theorem

Find

$$\oint_C (2x-y)dx + (x+3y)dy$$
$$={\iint }_R(3-2)dA={\iint }_{R}1dA=\text{Area(R)}$$

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9. Common GATE Question Types

Type 1: Find divergence/curl

Most frequent.

Type 2: Conservative vector fields

Condition:

$$\mathrm{\nabla }\times F=0$$

Type 3: Integration using Green’s theorem

Type 4: Work done by a force field

Type 5: Verify identities

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10. Common Mistakes to Avoid

1. Mixing up scalar and vector results (div gives scalar, curl gives vector).

2. Wrong order in determinant while computing curl.

3. Forgetting counter-clockwise orientation for Green’s Theorem.

4. Misapplying line integral parameterization.

5. Not checking boundary conditions.

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11. Short Revision Table

Concept Operator Output Meaning Gradient ∇f Vector     Maximum rate of change Divergence ∇⋅F Scalar     Outflow of field Curl ∇×F Vector     Rotation of field Line Integral                ∫C​F⋅dr Scalar     Work done Green’s Theorem Pdx+Qdy=∬(∂Q/∂y−∂P/∂x)                —     Area relation

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12. Final Summary (Exam-Ready)

• Gradient gives direction of steepest ascent.

• Divergence gives source/sink strength.

• Curl gives rotation of field.

• Line integrals measure work, circulation, or flux along a curve.

• Green’s theorem converts line integral → double integral.

• Vector identity formulas are extremely important for GATE.

• Conservative fields have zero curl.

• Area can be computed using:

$$\oint xdy=\text{Area}$$