Vector and Tensor Analysis MCQs

\[ \textbf{MCQs on Vector and Tensor Analysis with Answers} \] \[ \textbf{Q1: The divergence of the vector field } \vec{F} = \nabla \cdot \vec{F} \text{ is:} \] \[ \text{(A) } \text{A scalar quantity} \quad \text{(B) } \text{A vector quantity} \quad \text{(C) } \text{A tensor quantity} \quad \text{(D) } \text{None of these} \] \[ \textbf{Answer: (A) } \text{A scalar quantity} \] \[ \textbf{Q2: The curl of a vector field } \vec{F} \text{ is given by:} \] \[ \text{(A) } \nabla \cdot \vec{F} \quad \text{(B) } \nabla \times \vec{F} \quad \text{(C) } |\nabla| \vec{F} \quad \text{(D) } \vec{F} \cdot \nabla \] \[ \textbf{Answer: (B) } \nabla \times \vec{F} \] \[ \textbf{Q3: A tensor of rank 2 in three-dimensional space has how many components?} \] \[ \text{(A) } 6 \quad \text{(B) } 9 \quad \text{(C) } 3 \quad \text{(D) } 12 \] \[ \textbf{Answer: (B) } 9 \] \[ \textbf{Q4: The gradient of a scalar field is:} \] \[ \text{(A) } A scalar field \quad \text{(B) } A vector field \quad \text{(C) } A tensor field \quad \text{(D) } None of these \] \[ \textbf{Answer: (B) } A vector field \] \[ \textbf{Q5: Which of the following is true for a symmetric tensor?} \] \[ \text{(A) } T_{ij} = -T_{ji} \quad \text{(B) } T_{ij} = T_{ji} \quad \text{(C) } T_{ij} \neq T_{ji} \quad \text{(D) } T_{ij} = 0 \] \[ \textbf{Answer: (B) } T_{ij} = T_{ji} \] \[ \textbf{Q6: The determinant of the metric tensor in Cartesian coordinates is:} \] \[ \text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } -1 \quad \text{(D) } None of these \] \[ \textbf{Answer: (B) } 1 \] \[ \textbf{Q7: The dot product of two vectors results in:} \] \[ \text{(A) } A scalar quantity \quad \text{(B) } A vector quantity \quad \text{(C) } A tensor quantity \quad \text{(D) } None of these \] \[ \textbf{Answer: (A) } A scalar quantity \] \[ \textbf{Q8: A tensor that remains invariant under coordinate transformations is called:} \] \[ \text{(A) } A vector \quad \text{(B) } A scalar \quad \text{(C) } An isotropic tensor \quad \text{(D) } A covariant tensor \] \[ \textbf{Answer: (C) } An isotropic tensor \] \[ \textbf{Q9: The rank of the Kronecker delta } \delta_{ij} \text{ is:} \] \[ \text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } 2 \quad \text{(D) } None of these \] \[ \textbf{Answer: (C) } 2 \] \[ \textbf{Q10: The divergence theorem relates the surface integral of a vector field to:} \] \[ \text{(A) } Its curl \quad \text{(B) } Its gradient \quad \text{(C) } Its divergence \quad \text{(D) } None of these \] \[ \textbf{Answer: (C) } Its divergence \]