∇φ=(∂φ/∂x,∂φ/∂y,∂φ/∂z)=(yz,xz,xy). At (1,2,3) it becomes (2·3,1·3,1·2)=(6,3,2). Direction shows fastest rise.
For φ=x²y, the gradient at (1,2,0) equals
A (4,2,0)
B (2,1,0)
C (4,1,0)
D (2,2,0)
∇φ=(2xy, x², 0). At (1,2,0): (2·1·2, 1², 0)=(4,1,0). The z-component is zero because φ does not depend on z.
For φ=sin x + cos y + z, the gradient at (0,0,5) is
A (1,0,1)
B (0,1,1)
C (1,−1,1)
D (0,0,1)
∂φ/∂x=cos x, ∂φ/∂y=−sin y, ∂φ/∂z=1. At (0,0,5): (cos0,−sin0,1)=(1,0,1). It mixes x and z changes.
For scalar φ, the maximum directional derivative at a point equals
A ∇·φ
B |∇φ|
C |φ|
D ∇×φ
Directional derivative in unit direction u is ∇φ·u. The maximum value occurs when u aligns with ∇φ, giving max = |∇φ|. This is the steepest slope magnitude.
If φ(x,y,z)=constant defines a surface, its normal unit vector is along
A ∇×∇φ
B ∇·∇φ
C ∇φ direction
D φ vector
On a level surface, tangent directions give zero change in φ, so ∇φ is perpendicular to all tangents. Therefore ∇φ points normal to the surface at each point.
For F=(x², y², z²), divergence is
A 2x+2y+2z
B x+y+z
C x²+y²+z²
D 2(x+y+z)²
∇·F=∂(x²)/∂x + ∂(y²)/∂y + ∂(z²)/∂z =2x+2y+2z. It varies with position and indicates growing outflow with x,y,z.
For F=(xy, yz, zx), divergence is
A xy+yz+zx
B 0
C x+y+z
D 2(x+y+z)
∇·F=∂(xy)/∂x + ∂(yz)/∂y + ∂(zx)/∂z = y+z+x = x+y+z. Each term differentiates only with its own coordinate.
For F=(yz, zx, xy), divergence is
A 0
B x+y+z
C xy+yz+zx
D 2(x+y+z)
∂(yz)/∂x=0, ∂(zx)/∂y=0, ∂(xy)/∂z=0 since none depends on the differentiated variable. Hence ∇·F=0, making it solenoidal in Cartesian sense.
A solenoidal field guarantees which closed-surface result?
A Net work zero
B Net flux zero
C Net circulation zero
D Potential unique
If ∇·F=0 in a region, divergence theorem gives ∬closed F·n dS = ∭(∇·F)dV = 0. So total outward flux across any closed surface is zero.
For F=(0, x, 0), curl equals
A (0,0,−1)
B (1,0,0)
C (0,0,1)
D (0,1,0)
Curl k-component is ∂Q/∂x − ∂P/∂y = ∂(x)/∂x − 0 = 1. Other components are zero. So ∇×F=(0,0,1), rotation about z-axis.
For F=(0, 0, x), curl equals
A (0,−1,0)
B (0,1,0)
C (1,0,0)
D (0,0,1)
∇×F has j-component ∂P/∂z − ∂R/∂x = 0 − 1 = −1? Wait carefully: P=0, R=x. Then j = ∂P/∂z − ∂R/∂x = 0 − 1 = −1, so (0,−1,0).
Correct curl for F=(0, 0, x) is
A (0,1,0)
B (1,0,0)
C (0,0,0)
D (0,−1,0)
With F=(P,Q,R)=(0,0,x), curl is (∂R/∂y−∂Q/∂z, ∂P/∂z−∂R/∂x, ∂Q/∂x−∂P/∂y)=(0−0,0−1,0−0)=(0,−1,0).
For F=(y, z, x), divergence is
A 1
B 3
C 0
D x+y+z
∇·F=∂(y)/∂x + ∂(z)/∂y + ∂(x)/∂z = 0+0+0=0. Each component depends on a different variable, so each partial derivative becomes zero.
For F=(y, z, x), curl equals
A (1,1,1)
B (−1,−1,−1)
C (0,0,0)
D (1,−1,1)
Using curl formula: (∂R/∂y−∂Q/∂z, ∂P/∂z−∂R/∂x, ∂Q/∂x−∂P/∂y)=(0−1,0−1,0−1)=(−1,−1,−1).
Correct curl for F=(y, z, x) is
A (−1,−1,−1)
B (1,1,1)
C (0,0,0)
D (1,−1,1)
Here P=y, Q=z, R=x. Then ∂R/∂y=0, ∂Q/∂z=1 gives i=−1; ∂P/∂z=0, ∂R/∂x=1 gives j=−1; ∂Q/∂x=0, ∂P/∂y=1 gives k=−1.
If a field is conservative, which statement must hold in simply connected region?
A Divergence is zero
B Magnitude is zero
C Curl is zero
D Flux is maximum
A conservative field can be written as F=∇φ. Curl of a gradient is always zero, so ∇×F=0. Divergence need not be zero; it depends on φ.
If F=∇φ and φ is harmonic, then ∇·F equals
A 0
B 1
C |F|
D ∇×F
If F=∇φ, then ∇·F=∇·(∇φ)=∇²φ. Harmonic means ∇²φ=0, so divergence of F becomes zero in that region.
Work done by conservative force from A to B equals
A Path length
B Curl integral
C Potential drop
D Surface flux
For conservative force F=−∇V, work from A to B is W=∫F·dr = V(A)−V(B). It depends only on endpoints, not the chosen path.
In Stokes theorem, the surface normal direction affects
A Magnitude of field
B Sign of circulation
C Existence of curl
D Divergence value
Stokes theorem uses an oriented surface: boundary direction follows right-hand rule with surface normal. Reversing normal reverses boundary orientation, changing the sign of both integrals.
Gauss theorem requires the surface to be
A Open surface
B Plane curve
C Straight line
D Closed surface
Divergence theorem relates flux through a closed surface to volume integral of divergence inside. The boundary must enclose a volume; open surfaces do not define a unique enclosed volume.
For φ=ln r in 2D plane, ∇φ points
A Tangentially
B Downward only
C Radially outward
D Randomly
φ=ln r depends only on distance r from origin. Its gradient points in direction of increasing r, i.e., radially outward, with magnitude 1/r in plane polar form.
For F=(−y/(x²+y²), x/(x²+y²), 0), curl is
A Zero away origin
B Always 1
C Always −1
D Undefined everywhere
This field is like angular unit circulation field. In regions excluding the origin, curl is zero, but around the origin circulation is nonzero due to singularity, so not conservative globally.
A necessary condition for potential function existence is
A Closed surface only
B Simply connected region
C Nonzero divergence
D Constant magnitude
Curl-free is not enough if the region has holes. In a simply connected region, curl-free implies existence of potential. Otherwise, singularities or holes can produce nonzero loop integrals.
If ∮C F·dr ≠ 0 for some closed C, then F is
A Not conservative
B Always solenoidal
C Always harmonic
D Pure gradient
Conservative fields have zero circulation around every closed loop. If any closed path gives a nonzero line integral, then path independence fails, so the field cannot be conservative.
In cylindrical coordinates, divergence formula includes term
A r∂Fr/∂r
B (1/r²)∂Fr/∂r
C (1/r)∂(rFr)/∂r
D ∂Fr/∂θ only
Cylindrical divergence is ∇·F = (1/r)∂(rFr)/∂r + (1/r)∂Fθ/∂θ + ∂Fz/∂z. The (1/r) factor arises from coordinate scale.
In spherical coordinates, divergence includes radial term
A (1/r)∂(rFr)/∂r
B r²∂Fr/∂r
C ∂Fr/∂θ only
D (1/r²)∂(r²Fr)/∂r
Spherical divergence uses area scaling r². The term (1/r²)∂(r²Fr)/∂r accounts for changing surface area with r, plus angular terms involving sinθ.
If ∇×F=0 and ∇·F=0, then F can be written as
A Gradient of harmonic
B Curl of gradient
C Divergence of curl
D Constant only
If curl is zero, F=∇φ in simply connected region. If also divergence is zero, then ∇·F=∇²φ=0, so φ is harmonic. Thus F is gradient of harmonic potential.
For φ=x²−y², the gradient at (1,1,0) is
A (2,2,0)
B (−2,2,0)
C (2,−2,0)
D (0,0,0)
∇φ=(2x,−2y,0). At (1,1,0) it becomes (2,−2,0). This vector is normal to the level curve/surface and points toward fastest rise of φ.
For F=(x, −y, 0), divergence is
A 2
B 0
C x−y
D 1
∇·F=∂x/∂x + ∂(−y)/∂y + ∂0/∂z = 1 − 1 + 0 = 0. It indicates no net local source or sink overall.
∇²φ=∂²(xy)/∂x² + ∂²(xy)/∂y² + ∂²(xy)/∂z² = 0+0+0. Mixed terms vanish because second derivatives of xy in same variable are zero.
Laplacian of φ=sin x in 3D equals
A sin x
B −sin x
C 0
D −cos x
∇²φ=∂²(sin x)/∂x² + 0 + 0 = −sin x. Only x contributes. This shows Laplacian can return negative of the original function for sinusoidal fields.
In a potential flow (intro), fluid velocity often satisfies
A ∇·V ≠ 0
B V = constant only
C ∇×V = 0
D ∇²V = 1
Potential flow is irrotational, meaning curl of velocity is zero. Then velocity can be written as gradient of a velocity potential. Incompressibility adds ∇·V=0 additionally.
Flux density at a point is closely connected to
A Curl value
B Gradient value
C Jacobian only
D Divergence value
Divergence measures net outward flux per unit volume in the limit of small volume. That is essentially a local flux density measure, widely used in continuity and conservation equations.
Circulation density at a point is closely connected to
A Curl value
B Divergence value
C Laplacian value
D Potential value
Curl gives circulation per unit area in a limiting sense. It tells the axis and strength of local rotational tendency of the field, important in fluid rotation and electromagnetic relations.
If F has zero curl but region has a “hole”, then F may
A Become always constant
B Force divergence nonzero
C Fail to be conservative
D Make Laplacian undefined
Curl-free is not sufficient in regions that are not simply connected. A field can have zero curl everywhere yet have nonzero circulation around loops enclosing the hole, so no global potential exists.
For φ=x/y (y≠0), gradient components involve
A Quotient derivatives
B Only constants
C Only sines
D Only logs
φ=x/y depends on both x and y. Its gradient requires partial derivatives using quotient rule: ∂φ/∂x=1/y and ∂φ/∂y=−x/y², giving a nontrivial vector field.
For F=(∇φ), the line integral along any path from A to B equals
A φ(A)+φ(B)
B |∇φ|
C ∇²φ
D φ(B)−φ(A)
If F=∇φ, then by fundamental theorem for line integrals, ∫A→B F·dr = φ(B)−φ(A). The result is path independent, depending only on endpoints.
For F=−∇V, the work done by F from A to B equals
A V(B)−V(A)
B V(A)−V(B)
C 0 always
D ∇·F
With F=−∇V, line integral gives work W=∫F·dr = −(V(B)−V(A)) = V(A)−V(B). Work is positive when moving toward lower potential.
Jacobian is mainly used in vector calculus to
A Change variables
B Find curl only
C Find divergence only
D Remove integrals
Jacobian determinant appears when converting integrals between coordinate systems, like Cartesian to cylindrical or spherical. It adjusts volume/area elements so computed integrals remain correct under transformation.
In cylindrical coordinates, volume element dV equals
A dr dθ dz
B r² dr dθ dz
C r dr dθ dz
D dr dz
Cylindrical volume element includes factor r due to polar area scaling in xy-plane. So dV = r dr dθ dz. This comes from Jacobian of transformation (x=r cosθ, y=r sinθ).
In spherical coordinates, volume element dV equals
A r² sinθ dr dθ dφ
B r sinθ dr dθ dφ
C sinθ dr dθ dφ
D r² dr dθ dφ
Spherical volume element includes Jacobian r² sinθ. It accounts for radial stretching and angular scaling. Thus dV = r² sinθ dr dθ dφ, crucial for correct 3D integrals.
For F=(x, y, z), flux through sphere centered at origin is
A 4πR²
B 4πR³
C 0
D 2πR³
For sphere radius R, divergence ∇·F=3. Flux = ∭(∇·F)dV = 3·(4/3)πR³ = 4πR³. This uses divergence theorem and sphere volume.
For F=(x, y, z), flux through cube side a centered at origin is
A a²
B 0
C 3a³
D 6a³
Divergence ∇·F=3. Flux through closed cube equals ∭3 dV = 3·(cube volume). If cube side is a, volume is a³, so flux is 3a³.
For F=(−y, x, 0), divergence is
A 0
B 1
C 2
D −2
∇·F=∂(−y)/∂x + ∂(x)/∂y + ∂0/∂z = 0+0+0=0. It represents pure rotation field with no net source or sink.
For F=(−y, x, 0), circulation around circle radius R in xy-plane is
A vector field is irrotational if its circulation around small loops is
A Always maximum
B Approximately zero
C Always infinite
D Always negative
Curl measures circulation density. If curl is zero, then for sufficiently small loops, circulation becomes nearly zero. This reflects no local spinning tendency and supports existence of potential in proper regions.
A scalar potential is unique up to
A Additive constant
B Multiplicative constant
C Sign change only
D Rotation matrix
If ∇φ=F, then adding a constant C gives ∇(φ+C)=∇φ. So the potential is not unique; it can shift by any constant without changing the resulting vector field.
For conservative field F, potential difference between points equals
A Curl surface integral
B Divergence volume integral
C Negative line integral
D Gradient magnitude
If F=−∇V, then V(B)−V(A)=−∫A→B F·dr. Potential difference is the negative of work integral. This is the standard energy relation in conservative force fields.
The Laplacian link to divergence is expressed by
A ∇²φ = ∇×(∇φ)
B ∇²φ = ∇·(∇×F)
C ∇²φ = ∇×(∇·F)
D ∇²φ = ∇·(∇φ)
Laplacian of scalar φ is defined as divergence of its gradient. This operator sums second partial derivatives and appears naturally when combining gradient and divergence in physical field equations