Vector Fields

(requires JavaScript)

1. Sketch the vector field $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\left(x-y\right)\mathbf{i}+x\mathbf{j}$.
2. Sketch the vector field ${\displaystyle \mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\frac{y\mathbf{i}+x\mathbf{j}}{\sqrt{x^2+y^2}}}$.
3. Sketch the vector field ${\displaystyle \mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=-y\mathbf{k}}$.
4. Find the gradient vector field of $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\ln <!--FUNCTION\; APPLICATION-->\left(x+2y\right)$.
   ${\displaystyle \nabla f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\frac{\mathbf{i}}{x+2y}+\frac{2\mathbf{j}}{x+2y}}$
5. Find and sketch the gradient field of $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\frac{1}{4}{\left(x+y\right)}^2$.