Vector Calculus MCQs

  1. What is the gradient of a scalar function f(x,y,z)f(x, y, z)? A. A scalar quantity B. A vector perpendicular to f(x,y,z)f(x, y, z) C. A vector pointing in the direction of the greatest rate of increase of f(x,y,z)f(x, y, z) D. A scalar value that gives the maximum value of f(x,y,z)f(x, y, z) Answer: C
  2. If F=∇×A\mathbf{F} = \nabla \times \mathbf{A}, what type of vector field is F\mathbf{F}? A. Gradient Field B. Divergence-Free Field C. Conservative Field D. Radial Field Answer: B
  3. The divergence of a vector field F\mathbf{F} is defined as: A. ∇⋅F\nabla \cdot \mathbf{F} B. ∇×F\nabla \times \mathbf{F} C. ∇2F\nabla^2 \mathbf{F} D. None of the above Answer: A
  4. What does the curl of a vector field represent? A. Divergence of the field B. Circulation of the field per unit area C. Rate of change of a scalar field D. Laplacian of the field Answer: B
  5. Which theorem relates the line integral of a vector field around a closed curve to the surface integral of its curl? A. Green’s Theorem B. Gauss’s Divergence Theorem C. Stokes’ Theorem D. Fundamental Theorem of Calculus Answer: C
  6. The Laplacian operator ∇2\nabla^2 is applied to which type of function? A. Scalar functions only B. Vector functions only C. Both scalar and vector functions D. None of the above Answer: C
  7. The flux of a vector field F\mathbf{F} through a closed surface is given by: A. ∫S(∇⋅F) dV\int_S (\nabla \cdot \mathbf{F}) \, dV B. ∫S(F⋅n) dA\int_S (\mathbf{F} \cdot \mathbf{n}) \, dA C. ∫S(∇×F) dA\int_S (\nabla \times \mathbf{F}) \, dA D. None of the above Answer: B
  8. The line integral of F⋅dr\mathbf{F} \cdot d\mathbf{r} is independent of the path if F\mathbf{F} is: A. Solenoidal B. Irrotational C. Conservative D. Both B and C Answer: D
  9. In cylindrical coordinates, the differential element of volume is: A. dr dθ dzdr \, d\theta \, dz B. r dr dθ dzr \, dr \, d\theta \, dz C. r2 dr dθ dzr^2 \, dr \, d\theta \, dz D. r dr dz dθr \, dr \, dz \, d\theta Answer: B
  10. What is the physical interpretation of the divergence of a vector field? A. Rate of change of the field along a curve B. Curl of the field around a point C. Net rate of flow out of a point per unit volume D. Magnitude of the vector field at a point Answer: C

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