Lesson 1.4: Vector Calculus (Gradient, Divergence, Curl, Green’s & Stokes Theorem)
Vector Calculus is essential for mechanical, civil, and electrical engineering applications. GATE and PSU exams often test vector operations, physical interpretations, and fundamental theorems. Topics include Gradient, Divergence, Curl, Line/Surface/Volume Integrals, Green’s Theorem, and Stokes’ Theorem.
🔹 1. Gradient
Definition:
The gradient of a scalar function ϕ(x,y,z)\phi(x,y,z) is a vector pointing in the direction of maximum rate of increase.
∇⃗ϕ=∂ϕ∂xi^+∂ϕ∂yj^+∂ϕ∂zk^\vec{\nabla} \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}
Physical Meaning:
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Points in the direction of maximum increase of ϕ\phi
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Magnitude = rate of increase in that direction
Example:
ϕ=x2+y2+z2\phi = x^2 + y^2 + z^2 → ∇⃗ϕ=2xi^+2yj^+2zk^\vec{\nabla} \phi = 2x\hat{i} + 2y\hat{j} + 2z\hat{k}
🔹 2. Divergence
Definition:
Divergence of a vector field F⃗=Pi^+Qj^+Rk^\vec{F} = P\hat{i} + Q\hat{j} + R\hat{k} measures net outflow per unit volume:
∇⋅F⃗=∂P∂x+∂Q∂y+∂R∂z\nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
Applications:
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Fluid flow → source/sink strength
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Electrostatics → Gauss’s law
Example:
F⃗=xi^+yj^+zk^\vec{F} = x\hat{i} + y\hat{j} + z\hat{k} → ∇⋅F⃗=1+1+1=3\nabla \cdot \vec{F} = 1 + 1 + 1 = 3
🔹 3. Curl
Definition:
Curl measures rotation or circulation of a vector field:
∇×F⃗=∣i^j^k^∂∂x∂∂y∂∂zPQR∣\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}
Applications:
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Vorticity in fluid mechanics
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Electromagnetic field analysis
Example:
F⃗=−yi^+xj^+0k^\vec{F} = -y\hat{i} + x\hat{j} + 0\hat{k} → ∇×F⃗=0i^+0j^+2k^\nabla \times \vec{F} = 0\hat{i} + 0\hat{j} + 2\hat{k}
🔹 4. Line, Surface & Volume Integrals
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Line Integral: ∫CF⃗⋅dr⃗\int_C \vec{F} \cdot d\vec{r} → work done by force along path
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Surface Integral: ∬SF⃗⋅dS⃗\iint_S \vec{F} \cdot d\vec{S} → flux through surface
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Volume Integral: ∭Vf(x,y,z)dV\iiint_V f(x,y,z) dV → total quantity in volume
🔹 5. Green’s Theorem (2D)
Relates line integral around closed curve C to double integral over region R:
∮C(Pdx+Qdy)=∬R(∂Q∂x−∂P∂y)dA\oint_C (P dx + Q dy) = \iint_R \left( \frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y} \right) dA
Applications:
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Fluid flow circulation
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Area calculations
🔹 6. Stokes’ Theorem (3D)
Relates line integral over closed curve C to surface integral of curl over S:
∮CF⃗⋅dr⃗=∬S(∇×F⃗)⋅dS⃗\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}
Applications:
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Electromagnetics (Faraday’s law)
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Fluid mechanics → circulation
🔹 7. Solved Examples (PYQ Style)
Example 1 (GATE ME 2017):
F⃗=x2i^+y2j^\vec{F} = x^2 \hat{i} + y^2 \hat{j}. Compute divergence.
👉 Solution: ∇⋅F⃗=2x+2y\nabla \cdot \vec{F} = 2x + 2y
Example 2 (PSU Exam):
Verify Stokes’ theorem for F⃗=−yi^+xj^\vec{F} = -y\hat{i} + x\hat{j} over unit circle in xy-plane.
👉 Solution: LHS = RHS = 2π
🔹 8. Practice Exercises
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Compute ∇⃗ϕ\vec{\nabla} \phi for ϕ=xyz\phi = xyz.
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Find ∇⋅F⃗\nabla \cdot \vec{F} if F⃗=xi^−yj^+zk^\vec{F} = x\hat{i} – y\hat{j} + z\hat{k}.
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Compute curl of F⃗=yzi^+zxj^+xyk^\vec{F} = yz\hat{i} + zx\hat{j} + xy\hat{k}.
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Evaluate ∮CF⃗⋅dr⃗\oint_C \vec{F} \cdot d\vec{r} using Green’s theorem for F⃗=−yi^+xj^\vec{F} = -y\hat{i} + x\hat{j} around unit square.
🔹 9. Summary
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Gradient: Direction & rate of maximum increase of scalar function.
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Divergence: Net outflow of vector field (source/sink).
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Curl: Rotation of vector field (vorticity).
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Line/Surface/Volume Integrals: Work, flux, total quantity.
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Green’s & Stokes’ Theorems: Convert between line and surface integrals → simplifies calculations.