In vector calculus, the gradient of a scalar field gives what at a point?
A Net flux
B Circulation around loop
C Direction of max increase
D Total field value
The gradient ∇φ points in the direction where the scalar field φ increases most rapidly. Its magnitude equals the maximum rate of increase per unit distance at that point.
The operator “del” is written as which symbol?
A ∇
B Δ
C ⊗
D ∂
The del operator is denoted by ∇. It acts like a vector differential operator and is used to define gradient, divergence, curl, and Laplacian in vector calculus.
If φ(x,y,z) is scalar, then ∇φ is a
A Scalar
B Vector
C Matrix
D Constant
The gradient of a scalar field produces a vector field. It combines partial derivatives in x, y, and z directions, forming a vector that describes spatial change of φ.
The directional derivative of φ is maximum along
A Any tangent direction
B Unit normal only
C Gradient direction
D Zero-change direction
The directional derivative in direction u is ∇φ·u. This dot product is maximized when u is parallel to ∇φ, so maximum increase occurs along the gradient direction.
A level surface of φ(x,y,z) = constant has normal along
A ∇×φ
B ∇·φ
C φ itself
D ∇φ
On a level surface, φ is constant, so the change along the surface is zero. The gradient is perpendicular to the surface, so ∇φ gives the normal direction.
In Cartesian form, gradient of φ equals
A (∂φ/∂x, ∂φ/∂y, ∂φ/∂z)
B (∂x/∂φ, ∂y/∂φ, ∂z/∂φ)
C (φx, φy, φz) constants
D (φ, φ, φ)
In Cartesian coordinates, ∇φ = i(∂φ/∂x) + j(∂φ/∂y) + k(∂φ/∂z). It captures how φ changes in each coordinate direction.
If ∇φ = 0 everywhere in a region, φ is
A Always zero
B Infinite at boundary
C Constant in region
D A vector field
If all partial derivatives are zero, then φ has no spatial change anywhere in that region. So φ must be constant (could be any constant value, not necessarily zero).
If φ is temperature, ∇φ represents
A Heat generated
B Temperature slope vector
C Mass density
D Fluid velocity
For a temperature field, the gradient points toward fastest temperature increase. Its magnitude gives the maximum temperature change per unit distance, helping describe thermal variation in space.
A conservative field F can be written as
A F = ∇φ
B F = ∇×A
C F = ∇·G
D F = φA
A vector field is conservative if it is the gradient of some scalar potential φ. Then line integrals become path independent and depend only on endpoints.
In a conservative field, line integral depends on
A Path shape only
B Surface chosen
C Endpoints only
D Loop direction
For conservative fields, ∫F·dr is path independent. It equals φ(B) − φ(A), so only the start and end points matter, not the route taken.
Divergence of a vector field measures
A Rotational strength
B Potential energy
C Tangential change
D Outflow rate density
Divergence ∇·F gives net outward flux per unit volume near a point. Positive divergence suggests a source, negative suggests a sink, in physical interpretations like fluid flow.
Divergence of F is written as
A ∇F
B ∇×F
C ∇·F
D F·∇
Divergence is the dot product of del with a vector field: ∇·F. It produces a scalar field that summarizes local expansion or compression behavior of the vector field.
If ∇·F = 0 everywhere, the field is
A Solenoidal
B Conservative
C Irrotational
D Uniform only
A field with zero divergence is called solenoidal. It indicates no net source or sink anywhere locally, often modeling incompressible flow or magnetic fields in basic physics.
In incompressible fluid flow, velocity field V satisfies
A ∇×V = 0
B ∇·V = 0
C ∇V = 0
D ∇²V = 0
In incompressible flow, fluid density is constant and there is no local volume expansion. So the divergence of velocity is zero, expressing conservation of mass locally.
A positive divergence at a point indicates a
A Sink
B Vortex center
C Source
D Level surface
Positive divergence means net outward flow from a small volume around the point, like fluid being created or spreading outward. This is modeled as a source in vector field flow ideas.
Curl of a vector field measures
A Net outward flux
B Field magnitude only
C Potential value
D Local rotation tendency
Curl ∇×F measures how a vector field tends to rotate around a point. It relates to circulation per unit area and is used to describe vorticity in fluid motion.
Curl of F is written as
A ∇×F
B ∇·F
C ∇F
D F×F
Curl is defined as the cross product of del with a vector field: ∇×F. It produces a vector field indicating axis and strength of local rotation.
If ∇×F = 0 in a simply connected region, F is
A Solenoidal
B Always zero
C Irrotational
D Divergent only
Zero curl means no local rotation, so the field is irrotational. In a simply connected region, such a field can also be conservative and expressible as F = ∇φ.
A field that is both divergence-free and curl-free is
A Always constant
B Harmonic-type behavior
C Always radial
D Always circular
If ∇·F = 0 and ∇×F = 0, the field has no sources and no rotation. Under suitable conditions, it behaves like a potential field satisfying Laplace-type relations.
The Laplacian operator is written as
A ∇²
B ∇
C ∇·
D ∇×
The Laplacian is ∇², defined as ∇·(∇φ) for a scalar φ. It combines second partial derivatives and appears in heat, potential, and wave-type equations.
For scalar φ, Laplacian ∇²φ equals
A ∇×(∇φ)
B (∇φ)·(∇φ)
C ∇·(∇φ)
D ∇(∇·φ)
Laplacian of a scalar field is the divergence of its gradient. In Cartesian coordinates, it is ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z².
If φ satisfies ∇²φ = 0, φ is called
A Harmonic function
B Linear field
C Solenoidal field
D Rotational function
A scalar function with zero Laplacian is harmonic. Such functions arise in steady-state heat conduction and electrostatic potential in source-free regions.
Gradient is related to which type of field input?
A Scalar field
B Vector field
C Tensor field
D Random field
Gradient takes a scalar field φ and produces a vector field ∇φ. It describes how the scalar quantity changes in space, directionally and in magnitude.
Divergence is related to which type of field input?
A Scalar field
B Constant field only
C Vector field
D Surface only
Divergence operates on a vector field F and returns a scalar ∇·F. It measures local net outflow per unit volume, useful for flux and continuity ideas.
Curl is related to which type of field input?
A Scalar field
B Vector field
C Volume only
D Point only
Curl takes a vector field F and returns another vector field ∇×F. It describes local swirling behavior and is connected to circulation density.
The gradient of a constant scalar field is
A Zero vector
B Unit vector
C Random vector
D Divergence value
If φ is constant, all its partial derivatives are zero. Hence ∇φ = 0. This matches the idea that a constant field has no spatial change anywhere.
The divergence of a constant vector field is
A Zero vector
B Constant scalar
C Zero scalar
D Infinite scalar
A constant vector field has no change in any direction, so partial derivatives are zero. Divergence sums these derivatives, giving ∇·F = 0.
The curl of a constant vector field is
A Zero scalar
B Zero vector
C Constant scalar
D Random vector
Curl depends on spatial variation of vector components. If all components are constant, their derivatives vanish, so ∇×F becomes the zero vector.
The direction of steepest decrease of φ is
A ∇φ
B ∇·φ
C ∇×φ
D −∇φ
The gradient points to steepest increase. Therefore the steepest decrease is along the opposite direction, −∇φ, and has the same magnitude of maximum rate but negative sign.
Directional derivative of φ in unit direction u is
A ∇×u
B ∇·u
C ∇φ·u
D φ·u
The directional derivative measures rate of change of φ along u. It equals ∇φ·u, connecting the gradient vector with the chosen direction through a dot product.
Curl is most directly linked with which integral idea?
A Circulation integral
B Flux integral
C Volume average
D Endpoint difference
Curl relates to circulation density. Stokes’ theorem connects the surface integral of curl over a surface to the line integral (circulation) of the field around its boundary curve.
Divergence is most directly linked with which integral idea?
A Circulation around curve
B Tangent along curve
C Flux through surface
D Gradient on surface
Divergence connects to net flux leaving a volume. Gauss divergence theorem links the triple integral of divergence over a volume to the flux integral over the closed surface boundary.
A conservative field has which property for closed loop integral?
A Always positive
B Always zero
C Always maximum
D Always undefined
In a conservative field, the line integral around any closed path is zero because it depends only on endpoints. For a closed loop, start and end coincide, so result is zero.
If a vector field is conservative, it must be
A Divergence-free always
B Constant everywhere
C Purely radial
D Curl-free in region
In a simply connected region, a conservative field has zero curl. This condition helps test conservativeness; if curl is not zero, it cannot be a gradient field.
If F = ∇φ, then ∇×F equals
A ∇²φ
B ∇·F
C 0 vector
D φF
Curl of a gradient is always zero: ∇×(∇φ) = 0. This is a standard identity and expresses that gradient fields do not produce local rotation.
If F = ∇×A, then ∇·F equals
A 0 scalar
B 0 vector
C ∇²A
D ∇A
Divergence of a curl is always zero: ∇·(∇×A) = 0. This identity is widely used in physics, including magnetostatics where magnetic field is divergence-free.
The identity ∇·(∇×F) equals
A ∇²F
B 0 scalar
C 0 vector
D ∇(∇·F)
The divergence of any curl is always zero. This follows from cancellation of mixed partial derivatives under standard smoothness conditions, forming a key vector calculus identity.
The identity ∇×(∇φ) equals
A 0 scalar
B ∇²φ
C 0 vector
D ∇·φ
Curl of a gradient is zero because the field derived from a potential has no rotation. This property underlies why conservative fields are irrotational in suitable regions.
The identity ∇·(∇φ) is called
A Curl
B Gradient
C Divergence-free test
D Laplacian
∇·(∇φ) defines the Laplacian ∇²φ. In Cartesian coordinates, it becomes the sum of second derivatives and is important in potential and diffusion problems.
In electrostatics (basic), electric field E is often
A −∇V
B ∇×V
C ∇·V
D ∇²V
Electric field is the negative gradient of electric potential V in electrostatics: E = −∇V. The negative sign means the field points toward decreasing potential.
In fluid flow, vorticity is related to
A Divergence only
B Gradient of pressure
C Curl of velocity
D Laplacian of density
Vorticity measures local spinning of fluid elements and is defined as ω = ∇×V, where V is the velocity field. Higher vorticity means stronger rotational behavior.
A field with zero curl is also called
A Solenoidal
B Irrotational
C Divergent
D Compressible
Zero curl means no local rotation, so the field is irrotational. Under appropriate region conditions, it can be expressed as the gradient of a scalar potential.
A field with zero divergence is also called
A Solenoidal
B Irrotational
C Conservative
D Potential-only
A divergence-free field is solenoidal. It indicates no net “source” or “sink” at points, commonly used for incompressible velocity fields and magnetic fields.
The physical meaning of divergence in flow is
A Rotation strength
B Tangential speed
C Expansion/compression rate
D Pressure value
Divergence indicates how fluid expands or compresses locally. Positive divergence suggests local expansion (outflow), negative divergence suggests compression (inflow), linking to continuity ideas.
The physical meaning of curl in flow is
A Net mass outflow
B Density change
C Surface normal
D Local spin tendency
Curl indicates local rotational tendency of the flow. It relates to circulation per unit area, and in fluids it connects to vorticity, describing how fluid elements rotate around an axis.
Stokes’ theorem connects curl to
A Surface flux only
B Line circulation
C Volume integral
D Gradient change
Stokes’ theorem states that the surface integral of (∇×F)·n dS equals the line integral of F·dr around the boundary curve. It links curl to circulation.
Gauss divergence theorem connects divergence to
A Open curve length
B Curl circulation
C Closed surface flux
D Directional derivative
Gauss theorem states ∭(∇·F)dV equals ∬F·n dS over the closed boundary surface. It converts a volume integral of divergence into a surface flux integral.
A line integral of F·dr is mainly used to find
A Work along path
B Flux through surface
C Volume expansion
D Laplacian value
Line integrals measure accumulated effect along a curve, such as work done by a force field. For conservative fields, it equals potential difference; otherwise it depends on the path.
A surface integral of F·n dS represents
A Circulation density
B Directional derivative
C Flux through surface
D Potential change
The surface integral of F·n over a surface measures flux, meaning net flow crossing the surface. It is central to divergence theorem and physical flow/electric flux ideas.
A volume integral of ∇·F over a region equals
A Boundary circulation
B Boundary flux
C Gradient along curve
D Curl along line
By Gauss divergence theorem, ∭(∇·F)dV over a volume equals the net outward flux ∬F·n dS across the closed boundary surface of that volume.