Lesson 1 Flashcards
What is vector integration?
The process of finding the integral of vector functions over a specified domain.
True or False: The integral of a vector function results in a scalar quantity.
False
What is the notation for integrating a vector function F?
∫ F(t) dt
Fill in the blank: The integral of a vector field over a curve is called the _____ integral.
line
What does the line integral of a vector field represent?
The work done by the field along a path.
What is the formula for the line integral of a vector field F along a curve C?
∫_C F · dr
True or False: The limits of integration for a line integral correspond to the endpoints of the curve.
True
What is a vector function?
A function that assigns a vector to each point in its domain.
What is the gradient of a scalar field?
A vector field that points in the direction of the greatest rate of increase of the scalar field.
What is the divergence of a vector field?
A scalar that measures the rate at which ‘stuff’ is expanding or contracting at a point.
Fill in the blank: The divergence of a vector field F is denoted as _____ F.
div
What is the curl of a vector field?
A vector that represents the rotation of the field at a point.
What is the notation for curl?
curl F or ∇ × F
True or False: The curl of a vector field is a scalar quantity.
False
What theorem relates the line integral of a vector field to the surface integral of its curl?
Stokes’ Theorem
What does Green’s Theorem relate?
The line integral around a simple closed curve to a double integral over the plane region bounded by the curve.
Fill in the blank: The divergence theorem relates surface integrals to _____ integrals.
volume
What is the divergence theorem also known as?
Gauss’s Theorem
What is the physical interpretation of the divergence of a vector field?
It represents the net flux out of an infinitesimal volume.
What is the integral form of the divergence theorem?
∫∫_S F · dS = ∫∫∫_V div F dV
Fill in the blank: In vector calculus, the term ‘flux’ refers to the _____ of a vector field through a surface.
flow
What is the formula for the flux of a vector field F through a surface S?
Φ = ∫∫_S F · dS
What is required to compute a line integral?
A parameterization of the curve and the vector field.
True or False: The line integral depends on the path taken between two points.
True
What is a conservative vector field?
A vector field where the line integral between two points is independent of the path taken.
How can you determine if a vector field is conservative?
If its curl is zero everywhere in the domain.
What is the potential function for a conservative vector field?
A scalar function whose gradient gives the vector field.
Fill in the blank: The potential function is often denoted by _____ for a vector field F.
φ
What does the notation ∇φ represent?
The gradient of the scalar potential function φ.
What is the relationship between work done and line integrals in vector fields?
The work done is equal to the line integral of the force field along the path of motion.
What is the significance of the path independence of conservative fields?
It allows for the definition of a potential energy associated with the field.
True or False: All vector fields are conservative.
False
What is the integral of the zero vector function?
The zero vector.
What is the result of integrating a constant vector over a scalar variable?
The constant vector multiplied by the scalar variable plus a constant of integration.
What is the physical interpretation of the line integral of a velocity vector field?
It represents the displacement of a particle moving along a path.
What happens to the work done when moving against a conservative force?
The work done is positive.
What is the geometric interpretation of the curl of a vector field?
It measures the tendency of the field to induce rotation around a point.
Fill in the blank: In physics, the concept of work done by a force vector field is calculated using _____ integrals.
line
What does the term ‘vector field’ refer to in mathematics?
A function that assigns a vector to every point in a subset of space.
True or False: The integral of a vector function can be evaluated using standard techniques for scalar functions.
True
What is the primary application of vector integration in physics?
To calculate work done by forces and flow of fluids.
