What does the divergence of a vector field measure?

The amount of spread that the field has at a point.

True or False: A positive charge produces a vector field with negative divergence.

False. A positive charge produces a vector field with positive divergence.

The equation of continuity in fluid dynamics relates the rate of increase of mass to changes in what property?

Density

Riddle: I am a measure of how much a vector field curls around a point, often associated with fluid rotation. What am I?

Curl

Fill-in-the-blanks: The ______ connects the flux of a vector field with the volume integral of the divergence of the field.

Divergence theorem

Which theorem relates the surface integral of the curl of a vector field to the line integral around the boundary of an open surface?

Stoke's Theorem

True or False: The divergence of a solenoidal field is always non-zero.

False. The divergence of a solenoidal field is zero.

What is the divergence of the position vector in three-dimensional space?

3

Which coordinate system is used to describe the divergence and Laplacian in spherical coordinates?

Spherical polar coordinates

Fill-in-the-blanks: The Laplacian operator is defined as the ______ of the gradient operator.

Divergence

What does a conservative field imply about its curl?

The curl of a conservative field is zero.

Riddle: I vanish everywhere except at a single point, where I am infinitely high but my area remains one. What am I?

Dirac delta function

In fluid dynamics, what is the term for the measure of the tendency of a vector field to spread out or converge?

Divergence

True or False: The surface integral of a vector field can be calculated directly from its divergence.

True.

Fill-in-the-blanks: A vector field is said to be ______ if it can be expressed as the gradient of a scalar function.

Conservative

What is the significance of the divergence theorem in vector calculus?

It relates the surface integral of a vector field over a closed surface to the volume integral of its divergence.

Which mathematical operator acts on a scalar field to produce a vector field?

Gradient operator

Fill-in-the-blanks: The unit vectors in cylindrical coordinates are typically defined in terms of ______ and z.

Ρ and θ

True or False: The Laplacian operator can act on both scalar and vector fields.

True.

Riddle: I describe how much a field curls, but not how much it spreads. What am I?

Curl

What is the relationship between the divergence of a curl and the Laplacian operator?

The divergence of a curl is always zero.

Fill-in-the-blanks: Green's first identity relates the divergence of a vector field to a ______ over a volume.

Surface integral

What is required for a vector field to be uniquely specified in a volume defined by a closed surface?

Its divergence, curl, and normal component at all points on the surface.

Which theorem states that a vector field is uniquely defined by its divergence and curl in a given volume?

Uniqueness Theorem

What is the divergence theorem used for in vector calculus?

It connects the flux of a vector field through a closed surface to the volume integral of the divergence of the field.