37 Vector Calculus questions from the Engineering Mathematics bank, written for Indian campus drives and tech interviews. Every question has a verified answer and an AI-tutor explanation on placd.
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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
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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"?
Junior
A.Irrotational field
B.Green's theorem in the plane
C.Stokes' theorem
D.Gradient
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3. Which statement is correct?
Junior
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
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4. What is Divergence?
Junior
A.the scalar measuring the net outward flux per unit volume of a vector field at a point
B.converts the flux of a vector field through a closed surface into the volume integral of the divergence over the enclosed region
C.a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential
D.the vector measuring the circulation density, or local rotation, of a vector field at a point
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5. Which term means: "the scalar measuring the net outward flux per unit volume of a vector field at a point"?
Junior
A.Conservative field
B.Vanishing of divergence of a curl
C.Divergence
D.Stokes' theorem
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6. Which statement is correct?
Junior
A.Divergence — a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
B.Divergence — a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential
C.Divergence — a vector field whose divergence is identically zero, so it has no sources or sinks and its flux through any closed surface vanishes
D.Divergence — the scalar measuring the net outward flux per unit volume of a vector field at a point
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7. What is Curl?
Junior
A.the vector of partial derivatives of a scalar field, pointing in the direction of steepest increase with magnitude equal to that maximum rate
B.the identity stating that the divergence of the curl of any twice-differentiable vector field is identically zero
C.converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve
D.the vector measuring the circulation density, or local rotation, of a vector field at a point
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8. Which term means: "the vector measuring the circulation density, or local rotation, of a vector field at a point"?
Junior
A.Solenoidal field
B.Directional derivative
C.Curl
D.Green's theorem in the plane
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9. Which statement is correct?
Junior
A.Curl — the vector measuring the circulation density, or local rotation, of a vector field at a point
B.Curl — the vector of partial derivatives of a scalar field, pointing in the direction of steepest increase with magnitude equal to that maximum rate
C.Curl — a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
D.Curl — 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
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10. What is Directional derivative?
Junior
A.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.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 curl is identically zero, so it can be written as the gradient of a scalar potential
D.the scalar measuring the net outward flux per unit volume of a vector field at a point
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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"?
Junior
A.Gradient
B.Irrotational field
C.Green's theorem in the plane
D.Directional derivative
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12. Which statement is correct?
Junior
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
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13. What is Gauss divergence theorem?
Mid
A.the identity stating that the divergence of the curl of any twice-differentiable vector field is identically zero
B.the identity stating that the curl of the gradient of any twice-differentiable scalar field is the zero vector
C.converts the flux of a vector field through a closed surface into the volume integral of the divergence over the enclosed region
D.a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential
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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"?
Mid
A.Gauss divergence theorem
B.Solenoidal field
C.Gradient
D.Directional derivative
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15. Which statement is correct?
Mid
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
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16. What is Stokes' theorem?
Mid
A.the vector of partial derivatives of a scalar field, pointing in the direction of steepest increase with magnitude equal to that maximum rate
B.the identity stating that the curl of the gradient of any twice-differentiable scalar field is the zero vector
C.converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve
D.a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
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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"?
Mid
A.Vanishing of divergence of a curl
B.Stokes' theorem
C.Green's theorem in the plane
D.Irrotational field
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18. Which statement is correct?
Mid
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
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19. What is Green's theorem in the plane?
Mid
A.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.converts a line integral around a simple closed plane curve into a double integral over the region it encloses
C.converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve
D.the identity stating that the curl of the gradient of any twice-differentiable scalar field is the zero vector
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20. Which term means: "converts a line integral around a simple closed plane curve into a double integral over the region it encloses"?
Mid
A.Vanishing of curl of a gradient
B.Irrotational field
C.Conservative field
D.Green's theorem in the plane
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21. Which statement is correct?
Mid
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
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22. What is Solenoidal field?
Mid
A.a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
B.a vector field whose divergence is identically zero, so it has no sources or sinks and its flux through any closed surface vanishes
C.the scalar measuring the net outward flux per unit volume of a vector field at a point
D.converts a line integral around a simple closed plane curve into a double integral over the region it encloses
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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"?
Mid
A.Vanishing of curl of a gradient
B.Solenoidal field
C.Gradient
D.Directional derivative
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24. Which statement is correct?
Mid
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
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25. What is Irrotational field?
Mid
A.a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential
B.the scalar measuring the net outward flux per unit volume of a vector field at a point
C.converts the circulation of a vector field around a closed curve into the flux of the curl through any surface bounded by that curve
D.converts a line integral around a simple closed plane curve into a double integral over the region it encloses
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26. Which term means: "a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential"?
Mid
A.Irrotational field
B.Stokes' theorem
C.Green's theorem in the plane
D.Directional derivative
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27. Which statement is correct?
Mid
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
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28. What is Conservative field?
Senior
A.a vector field whose curl is identically zero, so it can be written as the gradient of a scalar potential
B.the identity stating that the curl of the gradient of any twice-differentiable scalar field is the zero vector
C.a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes
D.the scalar measuring the net outward flux per unit volume of a vector field at a point
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29. Which term means: "a vector field whose line integral is path independent, equivalently one whose closed-loop circulation always vanishes"?
Senior
A.Conservative field
B.Vanishing of curl of a gradient
C.Irrotational field
D.Solenoidal field
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30. Which statement is correct?
Senior
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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