Vector Calculus: Matching Vector Field Properties

This solution explains the matching between mathematical conditions and properties of vector fields ($\vec{A}$) or scalar functions ($\phi$), focusing on core concepts in vector calculus.

Matching List-I with List-II

We analyze each item in List-I and determine its corresponding property in List-II:

• A. $\vec{\nabla}\cdot\vec{A}=0$: This condition states that the divergence of the vector field $\vec{A}$ is zero. A vector field with zero divergence is called solenoidal.
  Therefore, A matches with III. $\vec{A}$ is solenoidal.
• B. $\oint \vec{A}\cdot d\vec{r}=0$: This condition signifies that the line integral of the vector field $\vec{A}$ around any closed path is zero. This is a fundamental property defining a conservative vector field.
  Therefore, B matches with I. $\vec{A}$ is conservative.
• C. $\vec{A} = \vec{\nabla}\phi$: This indicates that the vector field $\vec{A}$ can be expressed as the gradient of a scalar potential function $\phi$. By definition, the gradient $\vec{\nabla}\phi$ points in the direction of the maximum rate of increase of the function $\phi$. Also, fields derived from a scalar potential are conservative.
  Therefore, C matches with IV. $\vec{A}$ points in the direction of maximum increase of the function $\phi$.
• D. $\vec{\nabla}\times\vec{A}=0$: This condition states that the curl of the vector field $\vec{A}$ is zero. A vector field with zero curl is defined as being irrotational.
  Therefore, D matches with II. $\vec{A}$ is irrotational.

Correct Matching

Based on the analysis above, the correct matching is:

• A - III
• B - I
• C - IV
• D - II

This corresponds to Option D.