∇φ=(∂/∂x,∂/∂y,∂/∂z). Here ∂φ/∂x=2xy, ∂φ/∂y=x²+z², ∂φ/∂z=2yz. At (1,2,1) → (4,2,4).
Directional derivative of φ=x²+y² along u=(1/√2,1/√2) at (1,1) is
A A. √2
B C. 1/√2
C B. 2√2
D D. 2
∇φ=(2x,2y). At (1,1), ∇φ=(2,2). Directional derivative = ∇φ·u = (2,2)·(1/√2,1/√2)=4/√2=2√2.
If φ=xyz and u is unit vector along (1,−1,0), then u equals
A B. (1,−1,0)
B C. (−1/√2,1/√2,0)
C D. (0,0,1)
D A. (1/√2,−1/√2,0)
Unit vector is obtained by dividing by magnitude. |(1,−1,0)|=√2. So u=(1/√2,−1/√2,0). Using unit direction is required for directional derivative.
For φ=xyz, directional derivative along (1,−1,0) at (1,1,1) is
A B. √2
B A. 0
C C. −√2
D D. 1
∇φ=(yz,xz,xy). At (1,1,1): (1,1,1). Unit u=(1/√2,−1/√2,0). Dot: (1,1,1)·u = (1−1)/√2=0. No change along that direction.
For level surface φ=x²+y²+z²=9, unit normal at (2,1,2) is
A B. (2,1,2)
B C. (1/3,2/3,2/3)
C A. (2/3,1/3,2/3)
D D. (−2/3,−1/3,−2/3)
Normal is ∇φ=(2x,2y,2z)=(4,2,4). Unit normal is (4,2,4)/| (4,2,4) |. Magnitude=√(16+4+16)=6, so unit normal=(2/3,1/3,2/3).
For F=(x²y, yz, zx), divergence is
A B. x²+z+y
B C. 2xy+y+z
C D. xy+z+x
D A. 2xy+z+x
∇·F=∂(x²y)/∂x + ∂(yz)/∂y + ∂(zx)/∂z = 2xy + z + x. Each term differentiates only with its own coordinate.
For F=(xy, xz, yz), divergence is
A B. y+z+x
B C. 0
C A. x+y+z
D D. xy+yz+zx
∇·F=∂(xy)/∂x + ∂(xz)/∂y + ∂(yz)/∂z = y + 0 + y? Wait: ∂(yz)/∂z = y. So total becomes y + 0 + y = 2y. Not correct; compute carefully.
Correct divergence for F=(xy, xz, yz) is
A B. x+y+z
B A. 2y
C C. 2x
D D. 2z
Use ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z. P=xy → ∂/∂x=y. Q=xz → ∂/∂y=0. R=yz → ∂/∂z=y. Sum = y+0+y=2y.
If ∇·F is negative in a region, it suggests
A B. Source-like behavior
B C. Pure rotation only
C D. Constant potential
D A. Sink-like behavior
Negative divergence indicates net inward flow into a small volume, like a sink. It means more field lines enter than leave locally, common in compressing flows or inward flux regions.
For incompressible flow, continuity equation reduces to
A B. ∇×V=0
B C. ∇²V=0
C A. ∇·V=0
D D. ∇V=0
Incompressible means constant density, so mass conservation simplifies. The continuity equation becomes divergence of velocity zero, indicating no local volume expansion or compression of the fluid.
Compute: ∂R/∂y=∂(xy)/∂y=0, ∂Q/∂z=∂(zx)/∂z=x → i=0−x=−x. But ∂R/∂y is 0, yes. Then i=−x not zero. So option A is wrong. Let’s compute fully and correct.
Correct curl for F=(yz, zx, xy) is
A A. (−x, −y, −z)
B B. (x, y, z)
C C. (0,0,0)
D D. (x−y, y−z, z−x)
P=yz, Q=zx, R=xy. Curl = (∂R/∂y−∂Q/∂z, ∂P/∂z−∂R/∂x, ∂Q/∂x−∂P/∂y)=(0−x, y−y, z−z?) Wait: ∂P/∂z = y, ∂R/∂x = y → j=0. ∂Q/∂x=z, ∂P/∂y=z → k=0. So curl = (−x,0,0).
Final correct curl for F=(yz, zx, xy) is
A D. (0,0,0)
B A. (−x,0,0)
C B. (0,−y,0)
D C. (0,0,−z)
Using curl formula: i(∂R/∂y−∂Q/∂z)=i(0−x)=−xi. j(∂P/∂z−∂R/∂x)=j(y−y)=0. k(∂Q/∂x−∂P/∂y)=k(z−z)=0. So curl=(−x,0,0).
A field is irrotational if the circulation around every small loop is
A B. Extremely large
B C. Always negative
C A. Nearly zero
D D. Always fixed
Curl measures circulation density. If curl is zero, then circulation per unit area tends to zero for small loops, so loops have nearly zero circulation, indicating no local rotational tendency.
If ∇×F=0 but region is not simply connected, then F may be
A B. Always conservative
B A. Not globally conservative
C C. Always solenoidal
D D. Always constant
In regions with holes, curl-free fields can still have nonzero circulation around loops enclosing the hole. Hence a global potential may not exist even though local curl is zero.
For potential V, electric field is E=−∇V. If V increases in +x, E points
A B. +x direction
B C. +y direction
C D. zero direction
D A. −x direction
E=−∇V points toward decreasing potential. If V increases with x, gradient points +x, so E points opposite, i.e., toward −x direction.
If B=∇×A, then which statement is always true?
A B. ∇×B=0
B C. B=∇φ
C A. ∇·B=0
D D. |B| constant
Divergence of a curl is always zero: ∇·(∇×A)=0. So any field defined as curl of a vector potential is automatically solenoidal.
If F=∇φ, then which identity holds?
A B. ∇·F=0
B A. ∇×F=0
C C. ∇²F=0
D D. ∇F=0
Curl of a gradient is always zero: ∇×(∇φ)=0. This expresses that potential fields have no local rotation, a basic condition for conservative fields.
For φ=x²+y², Laplacian in 2D equals
A B. 2
B C. 0
C D. x+y
D A. 4
In 2D, ∇²φ=∂²/∂x²(x²)+∂²/∂y²(y²)=2+2=4. This constant indicates uniform curvature of the scalar field in the plane.
A harmonic scalar field satisfies
A A. ∇²φ=0
B B. ∇φ=0
C C. ∇×φ=0
D D. ∇·φ=0
Harmonic means Laplace equation holds: ∇²φ=0. Such fields arise in source-free steady problems like electrostatic potential in charge-free regions or steady heat distribution.
In cylindrical coordinates, the gradient has θ component proportional to
A B. r∂φ/∂θ
B C. ∂φ/∂θ only
C A. (1/r)∂φ/∂θ
D D. (1/r²)∂φ/∂θ
Because θ is an angle, physical distance in θ direction is r dθ. So derivative with respect to θ must be scaled by 1/r to convert angular rate into spatial rate.
In spherical coordinates, the θ component of gradient is proportional to
A B. (1/r²)∂φ/∂θ
B C. r∂φ/∂θ
C D. sinθ∂φ/∂θ
D A. (1/r)∂φ/∂θ
In spherical coordinates, moving in θ direction corresponds to arc length r dθ. Hence gradient θ component includes factor 1/r, converting angular change to spatial rate.
In spherical coordinates, azimuthal component of gradient is proportional to
A B. 1/(r cosθ)
B A. 1/(r sinθ)
C C. 1/sinθ
D D. 1/r²
Azimuthal angle φ produces arc length r sinθ dφ. Therefore gradient’s φ-component uses 1/(r sinθ) scaling to translate angular derivative into spatial derivative.
If ∇·F is constant k, then flux through closed surface equals
A B. k times area
B C. k times length
C A. k times volume
D D. always zero
By divergence theorem, ∬ F·n dS = ∭(∇·F)dV. If divergence is constant k, flux equals k∭dV = k(volume enclosed). Shape does not matter, only volume.
For F=(x,y,z), flux through sphere radius R equals
A A. 4πR³
B B. 4πR²
C C. 0
D D. 2πR³
∇·F=3. Flux = ∭3 dV = 3·(4/3)πR³ = 4πR³. This uses divergence theorem and the sphere volume.
For F=(0,0,z), flux through cube side a aligned axes equals
A B. a²
B C. 0
C D. 2a³
D A. a³
Divergence ∇·F=∂z/∂z=1. Flux through closed cube equals ∭1 dV = volume = a³. Net flux comes only from top and bottom faces.
For F=(0,0,z), flux through sphere radius R equals
A A. 0
B C. 4πR²
C B. (4/3)πR³
D D. 2πR³
∇·F=1. By divergence theorem, flux = ∭1 dV = volume of sphere = (4/3)πR³. Symmetry does not make it zero because divergence is positive everywhere.
For F=(−y,x,0), curl is
A B. (0,0,1)
B A. (0,0,2)
C C. (0,2,0)
D D. (2,0,0)
P=−y, Q=x. k-component: ∂Q/∂x − ∂P/∂y = 1 − (−1)=2. Other components are zero. This is a standard uniform rotation field about z-axis.
Using Stokes theorem, circulation of F=(−y,x,0) around circle radius R equals
A B. 2πR
B C. πR²
C D. 0
D A. 2πR²
Curl is (0,0,2). For a circle in xy-plane, surface normal is k. Circulation = ∬(curl·n)dS = ∬2 dS = 2(area) = 2·πR² = 2πR².
For scalar f and vector F, ∇·(fF) equals
A B. f∇×F
B C. ∇(f·F)
C A. ∇f·F + f∇·F
D D. 0 always
This is product rule for divergence. It splits divergence of scaled vector field into contribution from scalar variation (∇f·F) and from field divergence scaled by f.
For scalar f and vector F, ∇×(fF) equals
A A. ∇f×F + f∇×F
B B. ∇f·F + f∇·F
C C. ∇(f×F)
D D. 0 always
Curl product identity shows two contributions: rotation due to spatial change of scalar f (∇f×F) and rotation already present in F scaled by f. Useful in EM and fluids.
For scalars f,g, ∇(fg) equals
A B. ∇f·∇g
B C. fg∇
C D. f+g only
D A. f∇g + g∇f
Gradient product rule follows from partial derivative product rule in each coordinate. It helps expand gradients of multiplied fields, common in temperature-dependent conductivity and variable density models.
If F is conservative, then potential can be found by ensuring
A B. Curl maximum
B C. Divergence constant
C A. Mixed partial equality
D D. Flux always zero
When integrating components to find φ, consistency requires ∂²φ/∂x∂y = ∂²φ/∂y∂x etc. These mixed partial equalities match the curl-free condition and ensure a valid potential exists.
For F=(2xy, x², 0), a potential function is
A A. x²y + C
B B. 2x²y + C
C C. xy² + C
D D. x² + y² + C
If φ=x²y, then ∂φ/∂x=2xy and ∂φ/∂y=x², matching F components, with z-component zero. Hence F=∇φ with φ=x²y+C, confirming conservativeness.
For F=(y, x, 0), curl is
A B. (0,0,−2)
B C. (0,0,2)
C A. (0,0,0)
D D. (2,0,0)
P=y, Q=x. k-component: ∂Q/∂x − ∂P/∂y = 1 − 1 = 0. Other components are zero. So curl is zero, making it locally irrotational.
For F=(y, x, 0), a potential function can be chosen as
A B. x² + y² + C
B C. x/y + C
C D. x−y + C
D A. xy + C
Let φ=xy. Then ∂φ/∂x=y and ∂φ/∂y=x, matching F. Hence F=∇φ, so field is conservative and potential can be taken as φ=xy+C.
For line integral ∫C F·dr, if F=∇φ, then value equals
A B. Curve length
B A. Endpoint difference
C C. Surface flux
D D. Loop area
Fundamental theorem: ∫A→B ∇φ·dr = φ(B)−φ(A). It removes dependence on the curve C, reducing line integral to simple potential evaluation at endpoints.
For closed curve C and conservative F, ∮C F·dr equals
A B. 1
B C. 2π
C A. 0
D D. depends path
In a conservative field, line integral depends only on endpoints. For a closed curve, start and end are same, so integral is φ−φ=0. This is a key test for conservativeness.
For F=(x,0,0), flux through sphere radius R equals
A B. 0
B C. 4πR²
C D. 2πR³
D A. (4/3)πR³
Divergence ∇·F=1. Flux through closed sphere equals ∭1 dV = volume of sphere = (4/3)πR³. Symmetry does not cancel because divergence is positive.
For F=(x,0,0), circulation around circle in xy-plane is
A A. 0
B B. 2πR
C C. πR²
D D. 2πR²
On a circle in xy-plane, dr is tangential while F=(x,0,0) has x-direction component. Over a full loop, contributions cancel symmetrically, and since curl is zero, circulation is zero.
For F=(0,−z, y), divergence is
A B. 1
B A. 0
C C. −1
D D. y+z
∇·F=∂0/∂x + ∂(−z)/∂y + ∂(y)/∂z = 0+0+0=0. Each component depends on a different variable than the derivative variable.
For F=(0,−z, y), curl equals
A B. (−2,0,0)
B C. (0,2,0)
C D. (0,0,2)
D A. (2,0,0)
P=0, Q=−z, R=y. Curl i-component: ∂R/∂y − ∂Q/∂z = 1 − (−1)=2. Other components are zero. So curl=(2,0,0).
For F=(0,−z, y), Stokes theorem predicts circulation around circle normal to x-axis depends on
A B. Perimeter only
B C. Radius only
C A. Area of circle
D D. Endpoints only
Circulation equals ∬(curl·n)dS. If circle’s normal is along x-axis, curl·n equals 2. Hence circulation is 2 times the area, so it depends on area πR².
If ∇²φ is positive at a point, it often indicates
A B. Local rotation only
B A. Local source presence
C C. Zero gradient only
D D. Solenoidal behavior
Laplacian ∇²φ can be seen as divergence of gradient. In many physical models like potential theory, ∇²φ relates to source density. Positive value suggests net “source-like” behavior locally.
Laplacian in Cartesian coordinates for φ is
A B. φx+φy+φz
B C. φxy+φyz+φzx
C A. φxx+φyy+φzz
D D. φxxx only
In Cartesian form, ∇²φ equals sum of second partial derivatives: ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z². It measures overall curvature of scalar field.
If ∇×F is parallel to k-unit vector, the field mainly rotates about
A B. x-axis
B C. y-axis
C D. no axis
D A. z-axis
Curl direction gives axis of local rotation by right-hand rule. If curl points along k, rotation is about z-axis in xy-plane. Magnitude gives strength of this local spin tendency.
If ∇·F is constant zero and ∇×F is nonzero, then field is
A A. Solenoidal only
B B. Conservative only
C C. Both conservative
D D. Both zero
∇·F=0 means solenoidal. Nonzero curl means it is not irrotational, so not conservative in general. Such fields often represent pure rotational flows with no sources or sinks.
For F=(−y,x,0), field is
A B. Conservative, not solenoidal
B C. Both conservative
C A. Solenoidal, not conservative
D D. Neither property
Divergence is zero, so solenoidal. Curl is (0,0,2), not zero, so not conservative. It represents a rotational field with circulation around z-axis.
If scalar φ has gradient zero along a curve, then along that curve φ is
A B. Increasing
B C. Decreasing
C D. Undefined
D A. Constant
If directional derivative along curve tangent is zero everywhere, φ does not change as you move along that curve. Therefore φ stays constant along the curve, matching meaning of level curves/surfaces