Vector Calculus Flashcards
(25 cards)
What is the definition of a vector field?
A vector field is a function that assigns a vector to every point in a subset of space.
True or False: The gradient of a scalar field points in the direction of the steepest ascent.
True
Fill in the blank: The divergence of a vector field measures the __________ of the field at a point.
net rate of flow out of an infinitesimal volume
What is the symbol for the gradient operator?
∇ (nabla)
Multiple Choice: Which of the following operations results in a scalar field? (a) Divergence (b) Curl (c) Gradient
a) Divergence
What is the physical interpretation of curl in a vector field?
Curl measures the rotation or swirling of the field around a point.
True or False: The curl of a gradient is always zero.
True
What does the Laplacian operator represent in vector calculus?
The Laplacian operator represents the divergence of the gradient of a scalar field.
Fill in the blank: The notation for divergence is __________.
∇ · F
Multiple Choice: Which theorem relates surface integrals to volume integrals? (a) Green’s Theorem (b) Stokes’ Theorem (c) Divergence Theorem
c) Divergence Theorem
What is a conservative vector field?
A conservative vector field is one where the line integral between two points is independent of the path taken.
True or False: A vector field is conservative if its curl is zero.
True
What is Green’s Theorem used for?
Green’s Theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by the curve.
Fill in the blank: The integral of a vector field along a curve is called a __________ integral.
line
Multiple Choice: What does the term ‘flux’ refer to in vector calculus? (a) Volume of a vector (b) Flow of a vector field through a surface (c) Magnitude of a vector
b) Flow of a vector field through a surface
What is the relationship between the curl and circulation in a vector field?
The curl of a vector field at a point measures the tendency of the field to induce circulation around that point.
True or False: The divergence of a constant vector field is always zero.
True
What is the formula for the divergence of a vector field F = (P, Q, R)?
∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
Fill in the blank: The line integral of a vector field is defined as __________.
∫C F · dr
Multiple Choice: Which of the following is NOT a vector operation? (a) Dot product (b) Cross product (c) Scalar product
c) Scalar product
What is the divergence theorem also known as?
Gauss’s Theorem
True or False: The Laplacian of a function is a second-order differential operator.
True
What does Stokes’ Theorem relate?
Stokes’ Theorem relates a surface integral of the curl of a vector field over a surface to a line integral of the vector field over its boundary.
Fill in the blank: The vector identity ∇ × (∇ × F) = __________ is known as the curl of the curl.
∇(∇ · F) - ∇²F
