∇φ=(2x,2y,2z). Substituting (1,1,1) gives (2,2,2). Gradient points toward fastest increase of φ and its magnitude gives maximum rate of change at that point.
If φ has no change in direction u, then ∇φ·u equals
A 0
B 1
C −1
D ∞
Directional derivative in unit direction u is ∇φ·u. If φ does not change along u, the directional derivative is zero, meaning gradient is perpendicular to u.
For a level surface φ=constant, tangent directions satisfy
A ∇φ×t = 0
B ∇φ·t = 0
C ∇φ = t
D φ·t = 1
Along a level surface, φ stays constant, so change along any tangent direction t is zero. Hence directional derivative ∇φ·t must be zero, making ∇φ normal to the surface.
In cylindrical coordinates, the gradient includes scale factor
A r in z-term
B θ in r-term
C r in θ-term
D z in r-term
In cylindrical form, ∇φ has components involving (1/r)∂φ/∂θ. The 1/r factor appears because θ is an angular coordinate, so distances scale with r.
In spherical coordinates, the gradient includes scale factor
A 1/(r sinθ) in φ-term
B r sinθ in r-term
C 1/r in r-term
D sinθ in θ-term only
Spherical gradient uses unit vectors and scale factors: (1/r) for θ and 1/(r sinθ) for azimuthal angle φ. These factors convert angular change into spatial rate.
If F is conservative, a potential function exists such that
A F = ∇×A
B ∇·F = 0 always
C ∇×F ≠ 0
D F = −∇φ
Conservative fields can be written as gradient of a potential (often with a negative sign depending on convention). Then work done is path independent and equals potential difference between endpoints.
A common test for conservativeness in simply connected region is
A ∇×F = 0
B ∇·F = 0
C ∇F = 0
D ∇²F = 0
If curl of a vector field is zero everywhere in a simply connected region, the field is conservative. Then there exists a scalar potential φ with F = ∇φ.
For φ=x, the gradient in Cartesian is
A (0,1,0)
B (0,0,1)
C (1,0,0)
D (1,1,1)
∂φ/∂x=1, ∂φ/∂y=0, ∂φ/∂z=0, so ∇φ=(1,0,0). It points purely along the x-direction, matching the direction of increase of x.
If temperature field T has gradient pointing downward, it means
A T decreases downward
B T increases downward
C T constant downward
D Heat flux upward only
Gradient points toward maximum increase. If ∇T points downward, temperature increases fastest downward. Heat flow usually goes opposite to gradient (from hot to cold), but gradient direction is increase.
For vector field F=(x, y, z), divergence is
A 3
B 0
C 1
D 2
∇·F = ∂x/∂x + ∂y/∂y + ∂z/∂z = 1+1+1=3. This indicates uniform “source strength” everywhere for this simple outward field.
For F=(x,0,0), divergence is
A 0
B 2
C 1
D x
∇·F = ∂(x)/∂x + ∂0/∂y + ∂0/∂z = 1. It shows constant local expansion in x-direction, independent of point location.
For F=(0, y, 0), divergence is
A 0
B y
C 2
D 1
Divergence adds partial derivatives of each component with respect to its coordinate. Here ∂(y)/∂y=1 and other terms are zero, so divergence equals 1 everywhere.
Divergence at a point is best described as
A Work per length
B Flux per volume
C Rotation per length
D Gradient per area
Divergence measures net outward flux from a tiny closed surface divided by enclosed volume as volume shrinks. This gives a local flux density interpretation, common in fluid and field models.
A solenoidal field implies which statement is true?
A Net curl zero
B Potential exists always
C Net flux zero
D Field constant always
Solenoidal means ∇·F=0. By divergence theorem, net flux through any closed surface is zero (in suitable regions), meaning no net sources or sinks inside that volume.
For F=(−y, x, 0), the curl is
A (0,0,2)
B (0,0,0)
C (2,0,0)
D (0,2,0)
∇×F has k-component ∂Q/∂x − ∂P/∂y = ∂(x)/∂x − ∂(−y)/∂y = 1 − (−1)=2, others are zero. Rotation is about z-axis.
For F=(y,0,0), the curl is
A (0,0,0)
B (0,1,0)
C (0,0,−1)
D (1,0,0)
Curl k-component is ∂Q/∂x − ∂P/∂y = 0 − ∂(y)/∂y = −1. Other components vanish, so curl is (0,0,−1), indicating rotation about z-axis.
A vector field with zero curl in region is called
A Irrotational
B Solenoidal
C Divergent
D Turbulent
Irrotational means ∇×F=0, showing no local rotational tendency. In simply connected regions, such fields can often be written as gradient of a scalar potential.
Curl at a point is best described as
A Flux density
B Circulation density
C Potential gradient
D Work density
Curl relates to circulation per unit area. If you take a tiny loop around the point, the circulation divided by loop area approaches the component of curl along the loop’s normal direction.
A key condition for incompressible flow is
A ∇×V = 0
B ∇V = constant
C ∇·V = 0
D ∇²V = 0
In incompressible flow, density is constant and no local volume change occurs. Divergence of velocity becomes zero, expressing local mass conservation in differential form.
In electrostatics (intro), if charge density is zero, then
A ∇·E = 0
B ∇×E ≠ 0
C E = 0 always
D ∇²E = 1
Gauss’s law in differential form links divergence of electric field to charge density. In a charge-free region, divergence of E is zero, meaning no net electric sources locally.
In magnetostatics (basic), magnetic field B satisfies
A ∇×B = 0 always
B B = ∇φ only
C ∇²B = 0 always
D ∇·B = 0
Magnetic field has zero divergence, expressing absence of magnetic monopoles in standard classical theory. It means net magnetic flux through any closed surface is zero.
Stokes’ theorem relates a line integral to a
A Volume integral of div
B Line integral of grad
C Surface integral of curl
D Surface integral of grad
Stokes’ theorem states ∮C F·dr equals ∬S (∇×F)·n dS. It converts circulation around boundary curve into surface integral of curl over spanning surface.
Divergence theorem relates a surface flux to a
A Volume integral of div
B Line integral of F
C Surface curl integral
D Gradient integral only
Gauss divergence theorem: ∬S F·n dS = ∭V (∇·F) dV. It links total outward flux through closed surface to volume integral of divergence inside.
Green’s theorem (intro) relates a line integral in plane to
A Triple integral in space
B Double integral in region
C Surface integral in space
D Endpoint potential difference
Green’s theorem connects a line integral around a simple closed curve in the plane to a double integral over the enclosed region, linking circulation with partial derivatives in 2D.
If ∇²φ = 0, it commonly models
A Growing current flow
B Random turbulence
C Steady heat state
D Time-varying charge
Laplace equation ∇²φ=0 appears in steady-state situations with no internal sources, such as equilibrium temperature distribution or electrostatic potential in charge-free regions.
For φ=x², Laplacian in 3D equals
A 2
B 0
C 1
D 3
∇²φ = ∂²(x²)/∂x² + ∂²/∂y² + ∂²/∂z² = 2 + 0 + 0 = 2. Only x contributes nonzero second derivative.
For φ=x²+y², Laplacian in 3D equals
A 2
B 4
C 6
D 0
Second derivatives: ∂²/∂x² of x² is 2, ∂²/∂y² of y² is 2, and z term is 0. So ∇²φ=2+2+0=4.
For φ=x²+y²+z², Laplacian equals
A 2
B 3
C 6
D 9
∂²(x²)/∂x²=2, ∂²(y²)/∂y²=2, ∂²(z²)/∂z²=2. Summing gives ∇²φ=6, a constant everywhere in space.
The gradient of product identity is
A ∇(fg)=f∇g+g∇f
B ∇(fg)=∇f·∇g
C ∇(fg)=f+g
D ∇(fg)=0 always
Product rule for gradient works like single-variable calculus but vector form. For scalar f and g, ∇(fg)=f∇g+g∇f, useful in simplifying field expressions.
The divergence product identity for scalar f and vector F is
A ∇·(fF)=f∇×F
B ∇·(fF)=∇(f·F)
C ∇·(fF)=0 always
D ∇·(fF)=∇f·F+f∇·F
Divergence product rule expands using dot product: ∇·(fF) = (∇f)·F + f(∇·F). It’s used in conservation laws and manipulating flux forms.
The curl product identity for scalar f and vector F is
A ∇×(fF)=∇f·F
B ∇×(fF)=f∇·F
C ∇×(fF)=∇f×F+f∇×F
D ∇×(fF)=0 always
Curl product rule: ∇×(fF)=∇f×F + f(∇×F). It separates how scalar variation and vector rotation contribute to overall curl.
A conservative field implies existence of which function?
A Vector potential
B Scalar potential
C Flux function only
D Stream function only
Conservative fields are gradients of a scalar potential. This makes line integrals path independent and simplifies work calculations. In physics, many force fields derive from a potential energy function.
Finding potential φ from conservative F usually uses
A Integrate components
B Differentiate components
C Multiply components
D Take cross product
If F=∇φ, then φ can be found by integrating component relations: ∂φ/∂x=P, ∂φ/∂y=Q, ∂φ/∂z=R, ensuring consistency with mixed partial derivatives.
Circulation of F around closed curve C is written as
A ∬ F·n dS
B ∭ ∇·F dV
C ∇×F
D ∮ F·dr
Circulation measures total tangential effect along a closed path. It is the line integral ∮C F·dr. Stokes’ theorem connects it to surface integral of curl.
Flux of F through a surface S is written as
A ∮ F·dr
B ∭ F dV
C ∬ F·n dS
D ∇·F
Flux counts how much of the vector field passes through a surface. It equals ∬S F·n dS, where n is unit normal. For closed surfaces, divergence theorem applies.
The “source–sink” idea is mainly linked to
A Divergence
B Gradient
C Curl
D Laplacian only
Divergence describes local creation (source) or removal (sink) of flow. Positive divergence behaves like a source, negative like a sink, especially in fluid flow and field flux models.
A vector potential A is used when a field is written as
A F = ∇φ
B F = ∇×A
C F = ∇·A
D F = A·A
When a field is expressed as curl of another vector field, that other field is called a vector potential. This is common in magnetism where B can be written as ∇×A.
If F is solenoidal in a region, it is often expressible as
A F = ∇φ
B F = ∇(∇·A)
C F = ∇×A
D F = constant only
In suitable regions, divergence-free (solenoidal) fields can be represented as curl of a vector potential. This automatically guarantees zero divergence because divergence of a curl is always zero.
A key link between curl and conservative fields is
A ∇×F = 0
B ∇·F = 0
C ∇²F = 1
D ∇F = constant
Conservative fields have zero curl in simply connected regions. This indicates the field has no rotational tendency, allowing it to be expressed as a gradient of a potential function.
For F=(x²,0,0), divergence is
A 2x
B 0
C x²
D 2
∇·F = ∂(x²)/∂x + 0 + 0 = 2x. Divergence varies with x, meaning local outflow strength changes with position along x-axis.
For F=(0,0,z²), divergence is
A 0
B 2z
C z²
D 2
Divergence is ∂0/∂x + ∂0/∂y + ∂(z²)/∂z = 2z. It increases with z, indicating stronger “source” behavior as z grows.
For F=(yz,0,0), divergence is
A z
B y
C 0
D yz
∇·F = ∂(yz)/∂x + 0 + 0 = 0 because yz does not depend on x. So there is no local net outflow measured by divergence in this field.
For F=(0,xz,0), divergence is
A z
B x
C xz
D 0
∇·F = ∂0/∂x + ∂(xz)/∂y + ∂0/∂z = 0 because xz does not depend on y. Divergence is zero everywhere.
For F=(0,xy,0), divergence is
A x
B 0
C y
D xy
∇·F = ∂0/∂x + ∂(xy)/∂y + 0 = x. Wait—differentiate xy with respect to y gives x. So divergence equals x, not y.
For F=(0,xy,0), divergence is
A 0
B y
C x
D 2xy
Divergence is ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z. Here Fy=xy, so ∂Fy/∂y=x. Other derivatives are zero, so ∇·F = x.
Curl of gradient of any scalar field equals
A ∇²φ
B 0 vector
C ∇φ
D 1 vector
The identity ∇×(∇φ)=0 holds for smooth scalar fields. It shows gradient fields are irrotational, a key reason conservative fields have zero curl in proper regions.
Divergence of curl of any vector field equals
A 0 scalar
B ∇²F
C ∇·F
D ∇×F
The identity ∇·(∇×F)=0 always holds for smooth fields. Mixed partial derivatives cancel out, so any curl field is automatically divergence-free.
If a vector field has both ∇·F=0 and ∇×F=0, then
A Only sources present
B Only rotation present
C No sources, no rotation
D Always nonzero flux
Zero divergence means no sources or sinks, and zero curl means no local rotation. Such fields behave like very “smooth” fields, often linked to harmonic potentials in suitable domains.
Coordinate conversion basics: cylindrical radius r equals
A √(x²+y²)
B √(y²+z²)
C √(x²+z²)
D x+y
In cylindrical coordinates, r is the distance from the z-axis to the point in the xy-plane. Hence r = √(x²+y²). This helps connect Cartesian and cylindrical forms.
Coordinate conversion basics: spherical radius r equals
A √(x²+y²)
B x+y+z
C |x−y|
D √(x²+y²+z²)
In spherical coordinates, r is the distance from origin to the point in 3D space. So r = √(x²+y²+z²). It is the same as magnitude of position vector