Match the following : Codes :List - I List - II (a) Curl operator (i) Gradient (b) Del operator (ii) Volume to surface conversion (c) Divergence tdeorem (iii) Surface to line conversion (d) Stokes tdeorem (iv) Rotation
(a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)
Two operators and two theorems, each with one defining property.
(a) Curl → (iv) rotation. \(\nabla\times\overline{A}\) measures the circulation of a field per unit area — how much it swirls about a point. Its physical reading is literal: place a tiny paddle wheel in the field and the curl is what makes it spin. A field with zero curl everywhere is irrotational and can be written as the gradient of a scalar potential, which is exactly why an electrostatic field has a voltage.
(b) Del → (i) gradient. Applied to a scalar, the del operator produces the gradient \(\nabla V\) — a vector pointing in the direction of steepest increase, with magnitude equal to the rate of change in that direction. Del is of course also the operator behind divergence and curl, but those require a dot or a cross; standing alone in front of a scalar it is the gradient.
(c) Divergence theorem → (ii) volume to surface.
\(\oint_{S}\overline{A}\cdot d\overline{S}=\int_{V}\left(\nabla\cdot\overline{A}\right)dV\)
— a volume integral becomes a surface integral over the enclosing boundary. This is what turns the point form of Gauss's law into its familiar integral form.
(d) Stokes' theorem → (iii) surface to line.
\(\oint_{L}\overline{A}\cdot d\overline{l}=\int_{S}\left(\nabla\times\overline{A}\right)\cdot d\overline{S}\)
— a surface integral becomes a line integral round the bounding contour, which is how Ampere's and Faraday's laws pass between their differential and integral forms.
| Item | Meaning | Code |
|---|---|---|
| Curl | Rotation / circulation density | (iv) |
| Del on a scalar | Gradient | (i) |
| Divergence theorem | Volume → surface | (ii) |
| Stokes' theorem | Surface → line | (iii) |
The order (iv), (i), (ii), (iii) is option 1.
The pattern behind both theorems is the same idea in different dimensions: each converts an integral over a region into an integral over that region's boundary, one dimension lower. Divergence relates 3-D to 2-D and Stokes relates 2-D to 1-D; both are special cases of the generalised Stokes theorem.
Two identities that follow immediately and are worth memorising: \(\nabla\times\left(\nabla V\right)=0\) — a gradient field never curls — and \(\nabla\cdot\left(\nabla\times\overline{A}\right)=0\) — a curl field never diverges, which is precisely why \(\overline{B}=\nabla\times\overline{A}\) guarantees \(\nabla\cdot\overline{B}=0\).
Hence, the correct match is (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii).
Similar Questions
A scalar function V is given by V = 2xyz2. The gradient of V is given by:
Assertion (A) : Curl of any vector is a vector. It gives normal vector which is perpendicular to both the plane and the parent vector.
Reason (R) : The value of curl of H can be found by the expression :
(Curl H)Normal = \(\lim_{\Delta S\to0}\dfrac{\oint \vec{H}\cdot d\vec{l}}{\Delta S}\)
where $\Delta S$ is the planar area and $\overline{\text{dl}}$ is the line element.
Select your answer using the codes given below.
In cylindrical coordinates, the Laplace equation holds the following expression :
\(\nabla\times\nabla\times\vec{A}\) is equal to :
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