Vector Calculus: Gradient, Divergence, and Curl

Three operators, three different outputs

• Gradient $\nabla f$ takes a scalar field and returns a vector field pointing where $f$ increases fastest.
• Divergence $\nabla\cdot F$ takes a vector field and returns a scalar measuring how much the field spreads out from a point.
• Curl $\nabla\times F$ takes a vector field and returns a vector measuring its local rotation.

Worked example: gradient

For $f(x,y,z) = x^2yz^3$, $\nabla f = (2xyz^3,\; x^2z^3,\; 3x^2yz^2)$. At $(1,2,1)$ this is $(4, 1, 6)$.

Worked example: divergence and curl of a rotating field

For $F = (y, -x, 0)$: $\nabla\cdot F = 0 + 0 + 0 = 0$, and $\nabla\times F = (0-0,\; 0-0,\; -1-1) = (0,0,-2)$. Nothing flows in or out, and the field rotates about the $z$-axis, which is what the zero divergence and the $z$-directed curl say.

Worked example: a field that both spreads and turns

For $F = (xy,\; yz,\; zx)$, find $\nabla\cdot F$ and $\nabla\times F$ at $(1,2,3)$.

Worked example: the curl of a gradient

Using $\nabla f = (2xyz^3,\; x^2z^3,\; 3x^2yz^2)$ from above, the first component of its curl is $\dfrac{\partial}{\partial y}(3x^2yz^2) - \dfrac{\partial}{\partial z}(x^2z^3) = 3x^2z^2 - 3x^2z^2 = 0$, and the other two vanish the same way. In general $\nabla\times(\nabla f) = \vec{0}$ for any smooth $f$, a quick check on a computed gradient.

Common mistakes

• Writing divergence as a vector. It's a scalar; curl is the vector.
• Swapping the order inside a curl component. The first component is $\partial F_3/\partial y - \partial F_2/\partial z$, not the reverse.
• Evaluating before differentiating. Substitute the point only after taking the derivatives.
• Taking the gradient of a vector field. Gradient acts on a scalar function.

Where this comes up in exams

Gradient, divergence and curl are listed under Vector Analysis in the GATE 2027 EC syllabus and under Calculus in the ME syllabus; the EE syllabus lists vector identities, directional derivatives and the divergence and Stokes theorems.

Checked against GATE 2027 EC syllabus, GATE 2027 ME syllabus and GATE 2027 EE syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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