Vector fields, curl, and divergence

1 Vector fields, curl, and divergence

Find the curl and divergence of the vector field
\[ F(x,y) = \langle x, y \rangle . \]
Determine if the vector field is conservative or not. If it is, find a potential function.
Find the curl and divergence of the vector field
\[ F(x,y) = \langle y , x \rangle . \]
Determine if the vector field is conservative or not. If it is, find a potential function.
Find the curl and divergence of the vector field
\[ F(x,y) = \langle - y , x \rangle . \]
Determine if the vector field is conservative or not. If it is, find a potential function.
Find the curl and divergence of the vector field
\[ F(x,y) = \langle y e^{xy} , x e^{xy} \rangle . \]
Determine if the vector field is conservative or not. If it is, find a potential function.
Find the curl and divergence of the vector field
\[ F(x,y, z) = \langle 2 xyz , x^2z , x^2 yz \rangle . \]
Determine if the vector field is conservative or not. If it is, find a potential function.
Find the curl and divergence of the vector field
\[ F(x,y, z) = \langle yz^2 , xz^2 , 2xyz \rangle . \]
Determine if the vector field is conservative or not. If it is, find a potential function.
Find the curl and divergence of the vector field
\[ F(x,y, z) = \langle e^x + z , \sin z - y , x + y \cos z \rangle . \]
Determine if the vector field is conservative or not. If it is, find a potential function.
Find a vector field \(F(x,y)\) that has positive curl at the point \((0,0)\) and negative curl at the point \((10,0)\).
Find a vector field \(F(x,y)\) that has positive divergence at the point \((0,0)\) and negative divergence at the point \((0,10)\).
Find a vector field \(F(x,y)\) that has positive curl at the point \((1,1)\) and negative divergence at the point \((-1,-1)\).
Find a vector field \(F(x,y)\) with the property that:
  • The flow-line that starts at the point \((1,0)\) passes through the point \((2,0)\) at a later time.
  • The flow-line that starts at the point \((0,1)\) passes through the point \((0,2)\) at a later time.
Find a vector field \(F(x,y)\) with the property that:
  • The flow-line that starts at the point \((1,0)\) passes through the point \((2,0)\) at a later time.
  • The flow-line that starts at the point \((0,2)\) passes through the point \((0,1)\) at a later time.
Find a vector field \(F(x,y)\) with the property that:
  • The flow-line that starts at the point \((1,0)\) passes through the point \((0,1)\) at a later time.
  • The flow-line that starts at the point \((2,0)\) passes through the point \((0,2)\) at a later time.
Sketch a vector field \(F(x,y)\) with the property that:
  • The flow-line that starts at the point \((2,0)\) is at the point \((0,1)\) after one unit of time.
  • The flow-line that starts at the point \((0,2)\) is at the point \((1,0)\) after one unit of time.