Question

Write a formula for a two-dimensional vector field which has all vectors parallel to the xx-axis and all vectors on a horizontal line having the same magnitude

Answer 1

Answer:

$f(x,y)=u(y)\hat{i}$

Step-by-step explanation:

Let two- dimensional vector field

$f(x,y)=u(x,y)\hat{i}+v(x,y)\hat{j}$

We have to find the formula for a two-dimensional vector field which has all vectors parallel to the x-axis and all vectors on a horizontal line having the same magnitude.

When all  vector parallel to x- axis then

v(x,y)=0

Then, the vector field can be written as

$f(x,y)=u(x,y)\hat{i}$

When all vectors on a horizontal line having the same magnitude then

u(x,y)=u(y)

Substitute the values

Then, we get

$f(x,y)=u(y)\hat{i}$

Answer 2

The formula for a two-dimensional vector field will be:

F = F(x,y) = <u(x,y), v(x,y)> = u i + v j;

Where (i,j) are unit vectors in the coordinate directions

To make all vectors parallel to the x-axis
v(x,y)=0
F(x,y) = <u(x,y),0>

To make all vectors on a horizontal line having the same magnitude.
u(x,y) = u(y)
so
F(x,y) = <u(y),0> = u(y) i