Matilda Fox

2022-07-23

Why can ${v}_{x},{v}_{y}$ in $\mathbf{v}=(z,z,z)$ contain the z variable?

taguetzbo

Beginner2022-07-24Added 16 answers

A vector field assigns to every point in space a vector. This one takes a point in space and assigns the vector that is the z coordinate in all 3 directions. I would find it more natural for it to use angle brackets to indicate that the output is a vector and not a point, so

$v(x,y,z)=\u27e8z,z,z\u27e9$

For example, this would assign the vector $\u27e83,3,3\u27e9$ to every point on the plane z=3

$v(x,y,z)=\u27e8z,z,z\u27e9$

For example, this would assign the vector $\u27e83,3,3\u27e9$ to every point on the plane z=3

Ethen Frey

Beginner2022-07-25Added 6 answers

The velocities ${v}_{x}$ and ${v}_{y}$ are velocities that point in the x and y directions. They can also depend on depth, z. Think of velocities in a stream that are zero at the bottom of the river, and peak at the surface of the river. These components increase as the coordinate system elevation above the river bottom increases from 0 to $z=H$

Alternatively, think of a head wind to a horizontally flying aeroplane. That velocity depends on elevation above ground, but points (let's say) due North.

Alternatively, think of a head wind to a horizontally flying aeroplane. That velocity depends on elevation above ground, but points (let's say) due North.

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