37 real Vector Calculus questions from the Engineering Mathematics bank, as asked in Indian campus drives and tech interviews. Every question has a verified answer and an AI-tutor explanation on placd — free to start.
1. What is Gradient?
Junior
A.a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential
B.the vector of partial derivatives of a scalar field, pointing in the direction of steepest increase with magnitude equal to that maximum rate
C.a vector field whose divergence is identically zero, so it has no sources or sinks and its flux through any closed surface vanishes
D.a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
2. Which term means: "the vector of partial derivatives of a scalar field, pointing in the direction of steepest increase with magnitude equal to that maximum rate"?
A.Gradient — the rate of change of a scalar field along a specified unit direction, equal to the dot product of the gradient with that unit vector
B.Gradient — converts the flux of a vector field through a closed surface into the volume integral of the divergence over the enclosed region
C.Gradient — a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential
D.Gradient — the vector of partial derivatives of a scalar field, pointing in the direction of steepest increase with magnitude equal to that maximum rate
11. Which term means: "the rate of change of a scalar field along a specified unit direction, equal to the dot product of the gradient with that unit vector"?
A.Directional derivative — converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve
B.Directional derivative — the identity stating that the curl of the gradient of any twice-differentiable scalar field is the zero vector
C.Directional derivative — the rate of change of a scalar field along a specified unit direction, equal to the dot product of the gradient with that unit vector
D.Directional derivative — a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
14. Which term means: "converts the flux of a vector field through a closed surface into the volume integral of the divergence over the enclosed region"?
A.Gauss divergence theorem — the vector of partial derivatives of a scalar field, pointing in the direction of steepest increase with magnitude equal to that maximum rate
B.Gauss divergence theorem — converts the flux of a vector field through a closed surface into the volume integral of the divergence over the enclosed region
C.Gauss divergence theorem — a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
D.Gauss divergence theorem — the vector measuring the circulation density, or local rotation, of a vector field at a point
17. Which term means: "converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve"?
A.Stokes' theorem — the scalar measuring the net outward flux per unit volume of a vector field at a point
B.Stokes' theorem — converts the flux of a vector field through a closed surface into the volume integral of the divergence over the enclosed region
C.Stokes' theorem — the vector of partial derivatives of a scalar field, pointing in the direction of steepest increase with magnitude equal to that maximum rate
D.Stokes' theorem — converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve
A.Green's theorem in the plane — the vector measuring the circulation density, or local rotation, of a vector field at a point
B.Green's theorem in the plane — the identity stating that the divergence of the curl of any twice-differentiable vector field is identically zero
C.Green's theorem in the plane — the rate of change of a scalar field along a specified unit direction, equal to the dot product of the gradient with that unit vector
D.Green's theorem in the plane — converts a line integral around a simple closed plane curve into a double integral over the region it encloses
23. Which term means: "a vector field whose divergence is identically zero, so it has no sources or sinks and its flux through any closed surface vanishes"?
A.Solenoidal field — a vector field whose divergence is identically zero, so it has no sources or sinks and its flux through any closed surface vanishes
B.Solenoidal field — the scalar measuring the net outward flux per unit volume of a vector field at a point
C.Solenoidal field — converts a line integral around a simple closed plane curve into a double integral over the region it encloses
D.Solenoidal field — the identity stating that the curl of the gradient of any twice-differentiable scalar field is the zero vector
A.Irrotational field — converts a line integral around a simple closed plane curve into a double integral over the region it encloses
B.Irrotational field — converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve
C.Irrotational field — a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential
D.Irrotational field — a vector field whose divergence is identically zero, so it has no sources or sinks and its flux through any closed surface vanishes
A.Conservative field — converts a line integral around a simple closed plane curve into a double integral over the region it encloses
B.Conservative field — converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve
C.Conservative field — a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
D.Conservative field — converts the flux of a vector field through a closed surface into the volume integral of the divergence over the enclosed region
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