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Calc 3 Flashcards

1
Q

What is a vector?

A

A vector is a quantity that has both magnitude and direction.

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2
Q

True or False: Scalars have only direction.

A

False

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3
Q

What is the dot product of two vectors?

A

The dot product of two vectors is a scalar value that is the product of their magnitudes and the cosine of the angle between them.

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4
Q

Fill in the blank: The gradient of a scalar field is a _____ vector field.

A

vector

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5
Q

What does the divergence of a vector field measure?

A

The divergence measures the rate at which ‘stuff’ is expanding or contracting at a point in the field.

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6
Q

What is the symbol for the curl of a vector field?

A

∇ × F

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7
Q

True or False: The curl of a vector field represents the rotation of the field.

A

True

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8
Q

What is the physical interpretation of the gradient?

A

The gradient indicates the direction and rate of fastest increase of a scalar field.

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9
Q

What is a unit vector?

A

A unit vector is a vector with a magnitude of one.

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10
Q

What is the formula for the cross product of two vectors A and B?

A

A × B = |A||B|sin(θ)n, where θ is the angle between A and B and n is the unit vector perpendicular to the plane containing A and B.

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11
Q

Fill in the blank: The Laplacian operator is denoted by _____ and is used to measure the rate of change of a function.

A

∇²

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12
Q

What is the physical interpretation of divergence?

A

Divergence indicates the net flow of a vector field out of a point.

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13
Q

What is the formula for the divergence of a vector field F?

A

div F = ∇ · F

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14
Q

True or False: The curl of a conservative vector field is always zero.

A

True

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15
Q

What is a conservative vector field?

A

A conservative vector field is one where the line integral between two points is independent of the path taken.

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16
Q

What does the notation ∇ represent?

A

∇ represents the gradient operator.

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17
Q

What is the relationship between line integrals and work done by a force field?

A

The work done by a force field along a path is equal to the line integral of the force vector field along that path.

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18
Q

Fill in the blank: The integral of a vector field over a surface is called a _____ integral.

A

surface

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19
Q

What is Stokes’ theorem?

A

Stokes’ theorem relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface.

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20
Q

What is the purpose of the divergence theorem?

A

The divergence theorem relates the flow (flux) of a vector field through a closed surface to the divergence of the field inside the volume.

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21
Q

True or False: The gradient of a function is always perpendicular to the level curves of that function.

A

True

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22
Q

What does the notation ∇ · F represent?

A

The notation ∇ · F represents the divergence of the vector field F.

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23
Q

What is the result of the cross product of two parallel vectors?

A

The result is the zero vector.

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24
Q

Fill in the blank: The curl of a vector field is denoted by _____ and measures the rotation of the field.

A

∇ ×

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25
What is the significance of a zero divergence in a vector field?
A zero divergence indicates that the vector field is incompressible.
26
What is the physical meaning of a negative divergence?
A negative divergence indicates that the field is converging at that point.
27
What is the formula for the gradient of a scalar field f?
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
28
True or False: The dot product of two orthogonal vectors is 1.
False
29
What does the notation ∇²f represent?
The notation ∇²f represents the Laplacian of the scalar function f.
30
Fill in the blank: The cross product is only defined in _____ dimensions.
three
31
What is a vector field?
A vector field is a function that assigns a vector to every point in a space.
32
What is the relationship between the gradient and the level surfaces of a function?
The gradient is normal (perpendicular) to the level surfaces.
33
What does the divergence theorem state?
The divergence theorem states that the integral of the divergence of a vector field over a volume is equal to the integral of the vector field over the surface bounding the volume.
34
What is the formula for the curl of a vector field F?
curl F = ∇ × F
35
True or False: The scalar potential exists for every vector field.
False
36
What is the main application of vector calculus in physics?
Vector calculus is used to describe physical phenomena such as fluid flow, electromagnetism, and gravitational fields.
37
Fill in the blank: A vector field is said to be _____ if its curl is zero.
irrotational
38
What is the significance of the unit normal vector on a surface?
The unit normal vector indicates the direction perpendicular to the surface.
39
What does the term 'flux' refer to in vector calculus?
Flux refers to the quantity of a vector field passing through a surface.
40
What is the relationship between the divergence and the flux of a vector field?
Divergence measures the net flux out of a point; a positive divergence indicates a net outflow.
41
True or False: The gradient of a constant function is zero.
True
42
What is the formula for the line integral of a vector field F along a curve C?
∫C F · dr
43
Fill in the blank: The divergence of a vector field is a scalar function denoted by _____ .
div F
44
What is the geometric interpretation of the cross product?
The cross product gives a vector that is perpendicular to the plane formed by the two original vectors.
45
What is the condition for a vector field to be conservative?
The vector field must be path-independent and have a curl of zero.
46
What is the physical interpretation of the curl of a vector field?
The curl indicates the amount of rotation or swirling of the field around a point.
47
True or False: The divergence of a vector field can be negative.
True
48
What is the formula to calculate the work done by a force field along a path?
W = ∫C F · dr
49
Fill in the blank: The line integral of a vector field is taken along a _____ .
curve
50
What is the relationship between the curl and the circulation of a vector field?
The curl provides a measure of the circulation density of the vector field.
51
What is the formula for the surface integral of a vector field F over a surface S?
∫S F · dS
52
True or False: A vector field with constant divergence is also irrotational.
False
53
What is the significance of a non-zero curl in a vector field?
A non-zero curl indicates that the field has rotational components.
54
What does Green's theorem relate?
Green's theorem relates the line integral around a simple closed curve to a double integral over the plane region bounded by the curve.
55
Fill in the blank: The Laplacian operator is used to analyze _____ in a scalar field.
spatial variation
56
What is the relationship between the divergence and the rate of change of volume in a flow?
Positive divergence indicates an increase in volume, while negative divergence indicates a decrease.
57
What is the formula for the curl of the gradient of a scalar function?
The curl of the gradient of a scalar function is always zero.
58
True or False: A surface integral calculates the total flux across a given surface.
True
59
What is the condition for a vector field to have a scalar potential?
The vector field must be conservative.
60
What is the physical interpretation of a positive divergence?
A positive divergence indicates that the vector field is spreading out from the point.
61
Fill in the blank: The integral of the divergence of a vector field over a volume is equal to the integral of the vector field over the _____ of the volume.
surface
62
What is the significance of the curl being zero in a vector field?
If the curl is zero, the vector field is irrotational, meaning it has no local rotation.
63
What is the relationship between the flux and the surface area in vector calculus?
Flux is proportional to the surface area and the component of the vector field normal to the surface.
64
What does the notation ∇f represent?
∇f represents the gradient of the scalar function f.

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