For φ = x²y + y²z + z²x, the directional derivative at (1,1,1) along u=(1,1,1)/√3 is
A 10/√3
B 12/√3
C 9/√3
D 8/√3
∇φ=(2xy+z², x²+2yz, y²+2zx). At (1,1,1) → (3,3,3). Dot with u gives (3+3+3)/√3 = 9/√3.
Correct directional derivative value in Q1 is
A 10/√3
B 12/√3
C 8/√3
D 9/√3
At (1,1,1), ∇φ becomes (3,3,3). The unit direction u is (1,1,1)/√3. So D_u φ = ∇φ·u = (3,3,3)·(1,1,1)/√3 = 9/√3.
A necessary and sufficient condition for F to be conservative in simply connected region is
A ∇×F = 0
B ∇·F = 0
C ∇²F = 0
D F·dr = 0
In a simply connected region, curl-free implies existence of a scalar potential φ with F=∇φ. This gives path independence. Divergence-free alone does not guarantee conservativeness.
For F=(y/(x²+y²), −x/(x²+y²), 0), curl is zero in region x²+y²≠0, yet field is
A Always conservative
B Always solenoidal only
C Gradient of r
D Not globally conservative
The field has a singularity at the origin, so region excluding origin is not simply connected. Even with zero curl away from origin, circulation around loops enclosing origin is nonzero.
For the field in Q7, circulation around circle r=R in xy-plane is
A 2π
B 0
C −2π
D π
Parameterize circle: x=Rcos t, y=Rsin t. Field equals (sin t/R, −cos t/R,0). dr=(−R sin t, R cos t,0)dt. Dot gives −dt. Integral 0→2π gives −2π.
For F=(x, y, z), using divergence theorem, flux through surface of sphere radius R is
A 4πR³
B 4πR²
C (4/3)πR³
D 0
∇·F=3. Flux = ∭3 dV over sphere = 3·(4/3)πR³ = 4πR³. This avoids direct surface integration and is a standard divergence theorem application.
For F=(x, y, z), flux through closed cylinder of radius a, height h centered on z-axis is
A 2πa²h
B πa²h
C 0
D 3πa²h
Divergence is 3, so total flux through any closed surface equals 3 times enclosed volume. Cylinder volume is πa²h, hence flux = 3πa²h.
For vector identity, ∇·(∇×A) equals
A ∇²A
B 0
C ∇·A
D ∇×A
Divergence of a curl is always zero for smooth fields. This identity is heavily used in electromagnetism and ensures curl-generated fields are automatically divergence-free.
For identity, ∇×(∇φ) equals
A 0
B ∇²φ
C ∇φ
D ∇·φ
Curl of gradient is always zero. It shows gradient fields have no local rotation and supports the condition that conservative fields are irrotational in simply connected regions.
If φ satisfies ∇²φ = k (constant), then φ is
A Harmonic field
B Solenoidal field
C Irrotational field
D Poisson-type field
Laplace equation is ∇²φ=0 (harmonic). If ∇²φ equals a nonzero constant or source term, it becomes a Poisson equation, modeling presence of uniform source density.
For φ=x²+y²+z², evaluate ∇²φ
A 3
B 2
C 6
D 0
∇²φ=∂²/∂x²(x²)+∂²/∂y²(y²)+∂²/∂z²(z²)=2+2+2=6. This constant indicates uniform curvature of the field in all directions.
For F=(yz, zx, xy), divergence is
A 0
B x+y+z
C xy+yz+zx
D 1
∇·F=∂(yz)/∂x + ∂(zx)/∂y + ∂(xy)/∂z =0+0+0=0. Each component lacks dependence on the variable of differentiation.
For F=(yz, zx, xy), curl equals
A (−x,0,0)
B (z−y, x−z, y−x)
C (0,−y,0)
D (0,0,−z)
Curl = (∂R/∂y−∂Q/∂z, ∂P/∂z−∂R/∂x, ∂Q/∂x−∂P/∂y). Here: (x−x? wait) ∂R/∂y=x, ∂Q/∂z=x →0; ∂P/∂z=y, ∂R/∂x=y →0; ∂Q/∂x=z, ∂P/∂y=z →0. So curl is zero, not option D.
Correct curl for F=(yz, zx, xy) is
A (−x,0,0)
B (0,−y,0)
C (0,0,−z)
D (0,0,0)
Compute: ∂R/∂y=x and ∂Q/∂z=x so i=0. ∂P/∂z=y and ∂R/∂x=y so j=0. ∂Q/∂x=z and ∂P/∂y=z so k=0. Hence curl is zero.
If a nonzero vector field is both curl-free and divergence-free in simply connected region, then it can be written as
A ∇φ with ∇²φ=0
B ∇×A with ∇·A=0
C ∇²φ with φ=0
D ∇·F constant
Curl-free implies F=∇φ. Divergence-free then gives ∇·F=∇²φ=0, so φ is harmonic. Hence F is gradient of a harmonic potential in that region.
For scalar f and vector F, the identity ∇·(fF) contains term
A ∇f×F
B f∇×F
C f∇²F
D ∇f·F
Product rule: ∇·(fF)=∇f·F + f(∇·F). The first term accounts for how the scalar factor varies in space and contributes to net divergence.
For scalar f and vector F, the identity ∇×(fF) contains term
A ∇f·F
B ∇f×F
C f∇·F
D ∇²f
Curl product rule: ∇×(fF)=∇f×F + f(∇×F). The cross term arises from spatial variation of f and can create rotation even if F has zero curl.
In cylindrical coordinates, volume element for triple integral is
A r dr dθ dz
B dr dθ dz
C r² dr dθ dz
D dr dz
Jacobian for cylindrical transformation is r. So dV = r dr dθ dz. This factor accounts for increasing circle circumference with radius, ensuring correct volume computation.
In spherical coordinates, volume element for triple integral is
A r sinθ dr dθ dφ
B sinθ dr dθ dφ
C r² dr dθ dφ
D r² sinθ dr dθ dφ
Jacobian for spherical transformation is r² sinθ. It accounts for radial stretching and angular scaling. Using this factor is essential to compute correct volumes and integrals in spherical systems.
For F=(−y,x,0), divergence theorem predicts net flux through any closed surface is
A 2π
B 4π
C 0
D depends surface
∇·F=∂(−y)/∂x + ∂x/∂y + 0 = 0. Hence ∭(∇·F)dV=0 and so net flux ∬F·n dS is zero for any closed surface in regular region.
For F=(−y,x,0), Stokes theorem gives circulation around circle radius R equals
A 2πR²
B 2πR
C πR²
D 0
Curl is (0,0,2). For a circle in xy-plane, curl·n=2. Circulation equals ∬2 dS = 2(area)=2πR². Orientation may change sign, magnitude stays 2πR².
If ∇²φ = 0 in a region, then average value property implies φ at center equals
A Maximum on boundary
B Minimum on boundary
C Zero always
D Average on sphere
Harmonic functions satisfy mean value property: value at a point equals the average value over any sphere centered at that point (within the region). It is a deep property used in potential theory.
For F=(x²−y², 2xy, 0), the curl is
A (0,0,0)
B (0,0,4y)
C (0,0,0?)
D (0,0,4x)
P=x²−y², Q=2xy. k-component: ∂Q/∂x − ∂P/∂y = 2y − (−2y)=4y, not zero. So correct is (0,0,4y).
Correct curl for F=(x²−y², 2xy, 0) is
A (0,0,4x)
B (4y,0,0)
C (0,0,4y)
D (0,4y,0)
Curl k-component is ∂Q/∂x − ∂P/∂y. Here ∂Q/∂x=2y, ∂P/∂y=−2y, so difference is 4y. Other components are zero because R=0 and no z-dependence.
For F=(x²−y², 2xy, 0), divergence is
A 2x+2y
B 0
C 2x
D 2x+2x =4x
Divergence is ∂P/∂x + ∂Q/∂y + ∂R/∂z. Here ∂(x²−y²)/∂x=2x, ∂(2xy)/∂y=2x, total 4x.
A vector field with curl nonzero cannot be written as
A Pure gradient
B Curl of A
C Sum of fields
D Any vector field
Gradient fields always have zero curl. So if a field has nonzero curl somewhere in a simply connected region, it cannot be expressed purely as ∇φ there.
For F=(yz, zx, xy), since curl is zero and divergence is zero, the field is
A Not conservative anywhere
B Always rotational
C Conservative in simply connected region
D Only solenoidal
Curl-free implies conservative in simply connected region. Divergence-free adds that potential is harmonic. So in such regions, there exists φ with F=∇φ and ∇²φ=0.
Find potential φ for F=(yz, zx, xy) in simply connected region
A x+y+z + C
B x²y² + C
C xy+yz+zx + C
D xyz + C
If φ=xyz, then ∂φ/∂x=yz, ∂φ/∂y=xz, ∂φ/∂z=xy, matching F. Hence φ=xyz + C is a valid potential function.
For φ=xyz, Laplacian ∇²φ equals
A 0
B x+y+z
C 1
D xyz
Second derivatives: ∂²(xyz)/∂x²=0, similarly for y and z. Sum is 0. Hence φ=xyz is harmonic, consistent with F=∇φ being both curl-free and divergence-free.
For F=∇φ with φ harmonic, the flux through any closed surface is
A proportional area
B 0
C proportional length
D undefined
If φ harmonic, ∇²φ=0. Then ∇·(∇φ)=0. By divergence theorem, net flux of F=∇φ through any closed surface equals ∭0 dV=0.
For vector field F, Laplacian link (intro) often appears as
A ∇²F = ∇·(∇×F)
B ∇²F = ∇×(∇·F)
C ∇²F = 0 always
D ∇²F = ∇(∇·F) − ∇×(∇×F)
This is a standard vector identity relating vector Laplacian to divergence and curl. It is used in electromagnetism and fluid mechanics to rewrite differential equations in alternative forms.
For F=(x,y,z), curl equals
A (0,0,0)
B (1,1,1)
C (−1,−1,−1)
D (0,0,1)
Each component depends only on its own coordinate. Cross partial differences cancel: ∂z/∂y=0, ∂y/∂z=0 etc. So curl is zero, confirming it is conservative with potential φ=(x²+y²+z²)/2.
Potential for F=(x,y,z) can be chosen as
A x+y+z + C
B xyz + C
C (x²+y²+z²)/2 + C
D x²+y²+z² + C
If φ=(x²+y²+z²)/2, then ∇φ=(x,y,z). This matches F. The factor 1/2 ensures derivative of x² gives x, and similarly for y and z.
For F=(x,y,z), Laplacian of its potential φ equals
A 0
B 6
C 1
D 3
Potential is φ=(x²+y²+z²)/2. Laplacian is ∂²/∂x²(x²/2)=1, similarly for y and z. Sum gives 3. Thus ∇·F=3 matches ∇²φ.
For u being unit vector, maximum value of ∇φ·u equals
A 0
B |∇φ|
C |φ|
D ∇²φ
Dot product ∇φ·u is maximized when u is parallel to ∇φ. Then value equals |∇φ||u|=|∇φ| because |u|=1. This is steepest directional derivative.
For a curve C, if F is conservative, then ∫C F·dr equals
A φ(B)−φ(A)
B ∮C F·dr
C ∬S F·n dS
D ∭V ∇×F dV
Fundamental theorem for line integrals: if F=∇φ, then line integral from A to B equals φ(B)−φ(A) regardless of path. This is core idea of conservative fields.
For field F, if ∇×F is constant vector k, circulation around any small loop area A normal to k is approximately
A A/k
B k/A
C 0
D kA
Curl is circulation per unit area. For small planar loop with normal aligned to curl direction, circulation ≈ (curl·n)·Area. If curl magnitude is k and alignment is perfect, circulation ≈ kA.
For divergence constant c, flux through small closed surface enclosing volume V is approximately
A c/V
B V/c
C cV
D 0
Divergence is flux per unit volume. For small region where divergence ≈ c, net flux ≈ c times volume. This is local form of divergence theorem and helps physical interpretation.
For F=(−y/(x²+y²), x/(x²+y²), 0), divergence (away origin) is
A 1
B −1
C undefined
D 0
Away from origin, the field is purely tangential with no net radial outflow, giving zero divergence. The singularity at origin causes global issues, but locally (x²+y²≠0) divergence is zero.
For F=(−y/(x²+y²), x/(x²+y²), 0), field is best described as
A Pure source field
B Pure circulation field
C Pure gradient field
D Uniform radial field
The direction is tangential to circles centered at origin and magnitude ~1/r. It produces nonzero circulation around origin but no net flux outward, so it represents a circulation-type field.
For Laplacian in cylindrical coordinates (intro), a key extra term appears as
A (1/r)∂/∂r
B r∂/∂r
C (1/r²)∂/∂r only
D ∂/∂θ only
Cylindrical Laplacian includes (1/r)∂/∂r(r∂φ/∂r) plus (1/r²)∂²φ/∂θ² and ∂²φ/∂z². The (1/r) structure comes from radial geometry.
In spherical Laplacian (intro), angular part includes factor
A 1/r
B sinθ only
C r² only
D 1/(r² sinθ)
Spherical Laplacian has angular terms: (1/(r² sinθ))∂/∂θ(sinθ ∂φ/∂θ) and (1/(r² sin²θ))∂²φ/∂φ². These scale with r² and sinθ.
For F=(y,−x,0), circulation around circle radius R in xy-plane (counterclockwise) is
A −2πR²
B 2πR²
C 0
D −2πR
Curl is (0,0,−2). For CCW orientation, normal is +k. Circulation = ∬(curl·n)dS = ∬(−2)dS = −2·πR² = −2πR².
For F=(y,−x,0), divergence is
A 2
B −2
C x+y
D 0
∇·F=∂y/∂x + ∂(−x)/∂y + 0 = 0+0+0=0. It is a pure rotational field with no net source or sink.
A vector field with nonzero curl can still have zero divergence, meaning it is
A Source without rotation
B Rotation without source
C Both absent
D Both present always
Curl measures rotation and divergence measures source/sink strength. Many fields (like circular flow) rotate strongly but have no net outflow, giving nonzero curl but zero divergence.
If F is solenoidal and given by F=∇×A, then flux through any closed surface is
A constant nonzero
B depends on path
C infinite
D 0
If F=∇×A, then ∇·F=0 always. By divergence theorem, net flux through any closed surface equals ∭0 dV = 0. This is why magnetic flux through closed surfaces is zero.