vector calculus

When can a vector field be expressed as the gradient of a scalar field?

• When it has a non-zero curl
• When it has a non-zero divergence
• When it is both irrotational and solenoidal
• When it is constant throughout its domain

What does the divergence of a vector field represent?

• Rate at which the field spreads out from a given point
• Rate of change of the scalar field in a given direction
• Rate of change of the vector field
• Magnitude of the vector field

What is the significance of Stokes' theorem in vector calculus?

• It generalizes Green's theorem to higher dimensions
• It relates the gradient of a scalar field to the divergence of a vector field
• It explains the concept of solenoidal vector fields
• It represents the rate of change of a scalar field

How is the divergence of a vector field mathematically represented?

• ∇f = ∇⋅V
• ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k
• ∇⋅V = (∂Vx/∂x) + (∂Vy/∂y) + (∂Vz/∂z)
• ∇⋅V = ∇f

What is the physical interpretation of the gradient of a scalar field?

• Rate of change of the vector field
• Rate at which the field spreads out from a given point
• Direction of the steepest increase of the scalar field
• Magnitude of the scalar field

What is the physical interpretation of an irrotational vector field?

• Its divergence is zero everywhere
• It represents a swirling motion
• It has sources and sinks at every point
• It has no rotation or vorticity at any point

What does the gradient of a scalar field represent?

• Rate of change of the vector field
• Direction of the smallest change in the scalar field
• Magnitude of the scalar field
• Rate of change of the scalar field in a given direction

What does Green's theorem relate a line integral around a closed curve to?

• A double integral over the region enclosed by the curve
• The gradient of the vector field
• The divergence of the vector field
• The curl of the vector field

How is the gradient of a scalar field mathematically represented?

• ∇f = ∇⋅V
• ∇⋅V = (∂Vx/∂x) + (∂Vy/∂y) + (∂Vz/∂z)
• ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k
• ∇⋅V = ∇f

What does it mean for a vector field to be solenoidal?

• It has sources and sinks at every point
• It represents a swirling motion
• Its curl is zero everywhere
• Its divergence is zero everywhere