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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