Exercise 8

Assume that $x,y \geq 0$ for all of the vector fields in this question. Select an answer for each question and explain your reasoning.

1. Let $\vF_1= y \vec{i}$.

   1. The vector field $\vF_1$ is

      • parallel to the x-axis
      • parallel to the y-axis
      • neither

   2. As $x$ increases,

      • the length of the vector field increases
      • the length of the vector field decreases
      • neither

   3. As $y$ increases,

      • the length of the vector field increases
      • the length of the vector field decreases
      • neither

2. Let $\vF_2= \langle y, 1 \rangle$.

   1. The vector field $\vF_2$ is

      • parallel to the x-axis
      • parallel to the y-axis
      • neither

   2. As $x$ increases,

      • the length of the vector field increases
      • the length of the vector field decreases
      • neither

   3. As $y$ increases,

      • the length of the vector field increases
      • the length of the vector field decreases
      • neither

3. Let $\vF_3= (x+e^{1-y})\vec{j}$.

   1. The vector field $\vF_3$ is

      • parallel to the x-axis
      • parallel to the y-axis
      • neither

   2. As $x$ increases,

      • the length of the vector field increases
      • the length of the vector field decreases
      • neither

   3. As $y$ increases,

      • the length of the vector field increases
      • the length of the vector field decreases
      • neither

4. Let $\vF_4= \nabla(y^4+e^{2x})$.

   1. The vector field $\vF_4$ is

      • parallel to the x-axis
      • parallel to the y-axis
      • neither

   2. As $x$ increases,

      • the length of the vector field increases
      • the length of the vector field decreases
      • neither

   3. As $y$ increases,

      • the length of the vector field increases
      • the length of the vector field decreases
      • neither