Section 1
Vector Fields
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y)=\mathbf{i}+\frac{1}{2} \mathbf{j}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y)=2 \mathbf{i}-\mathbf{j}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y)=\mathbf{i}+\frac{1}{2} y \mathbf{j}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y)=x \mathbf{i}+\frac{1}{2} y \mathbf{j}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y)=-\frac{1}{2} \mathbf{i}+(y-x) \mathbf{j}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y)=y \mathbf{i}+(x+y) \mathbf{j}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y)=\frac{y \mathbf{i}+x \mathbf{j}}{\sqrt{x^2+y^2}}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y)=\frac{y \mathbf{i}-x \mathbf{j}}{\sqrt{x^2+y^2}}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y, z)=\mathbf{i}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y, z)=z \mathbf{i}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y, z)=-y \mathbf{i}$
Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.$\mathbf{F}(x, y, z)=\mathbf{i}+\mathbf{k}$
Match the vector fields $\mathbf{F}$ with the plots labeled I-VI. Give reasons for your choices.$\mathbf{F}(x, y)=\langle x,-y\rangle$
Figure I-VI can't copy
Match the vector fields $\mathbf{F}$ with the plots labeled I-VI. Give reasons for your choices.$\mathbf{F}(x, y)=\langle y, x-y\rangle$
Match the vector fields $\mathbf{F}$ with the plots labeled I-VI. Give reasons for your choices.$\mathbf{F}(x, y)=\langle y, y+2\rangle$
Match the vector fields $\mathbf{F}$ with the plots labeled I-VI. Give reasons for your choices.$\mathbf{F}(x, y)=\langle y, 2 x\rangle$
Match the vector fields $\mathbf{F}$ with the plots labeled I-VI. Give reasons for your choices.$\mathbf{F}(x, y)=\langle\sin y, \cos x\rangle$
Match the vector fields $\mathbf{F}$ with the plots labeled I-VI. Give reasons for your choices.$\mathbf{F}(x, y)=\langle\cos (x+y), x\rangle$
Match the vector fields $\mathbf{F}$ on $\mathbb{R}^3$ with the plots labeled I-IV. Give reasons for your choices.$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}$
Figure I-IV can't copy
Match the vector fields $\mathbf{F}$ on $\mathbb{R}^3$ with the plots labeled I-IV. Give reasons for your choices.$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+z \mathbf{k}$
Match the vector fields $\mathbf{F}$ on $\mathbb{R}^3$ with the plots labeled I-IV. Give reasons for your choices.$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+3 \mathbf{k}$
Match the vector fields $\mathbf{F}$ on $\mathbb{R}^3$ with the plots labeled I-IV. Give reasons for your choices.$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$
Use graphing software to plot the vector field
$$\mathbf{F}(x, y)=\left(y^2-2 x y\right) \mathbf{i}+\left(3 x y-6 x^2\right) \mathbf{j}$$
Explain the appearance by finding the set of points $(x, y)$ such that $\mathbf{F}(x, y)=\mathbf{0}$.
Let $\mathbf{F}(\mathbf{x})=\left(r^2-2 r\right) \mathbf{x}$, where $\mathbf{x}=\langle x, y\rangle$ and $r=|\mathbf{x}|$. Use graphing software to plot this vector field in various domains until you can see what is happening. Describe the appearance of the plot and explain it by finding the points where $\mathbf{F}(\mathbf{x})=\mathbf{0}$.
Find the gradient vector field $\nabla f$ of $f$.$f(x, y)=y \sin (x y)$
Find the gradient vector field $\nabla f$ of $f$.$f(s, t)=\sqrt{2 s+3 t}$
Find the gradient vector field $\nabla f$ of $f$.$f(x, y, z)=\sqrt{x^2+y^2+z^2}$
Find the gradient vector field $\nabla f$ of $f$.$f(x, y, z)=x^2 y e^{y / z}$
Find the gradient vector field $\nabla f$ of $f$ and sketch it.$f(x, y)=\frac{1}{2}(x-y)^2$
Find the gradient vector field $\nabla f$ of $f$ and sketch it.$f(x, y)=\frac{1}{2}\left(x^2-y^2\right)$
Match the functions $f$ with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices.$f(x, y)=x^2+y^2$
Match the functions $f$ with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices.$f(x, y)=x(x+y)$
Match the functions $f$ with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices.$f(x, y)=(x+y)^2$
Match the functions $f$ with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices.$f(x, y)=\sin \sqrt{x^2+y^2}$
Plot the gradient vector field of $f$ together with a contour map of $f$. Explain how they are related to each other.$f(x, y)=\ln \left(1+x^2+2 y^2\right)$
Plot the gradient vector field of $f$ together with a contour map of $f$. Explain how they are related to each other.$f(x, y)=\cos x-2 \sin y$
A particle moves in a velocity field $\mathbf{V}(x, y)=\left\langle x^2, x+y^2\right\rangle$. If it is at position $(2,1)$ at time $t=3$, estimate its location at time $t=3.01$.
At time $t=1$, a particle is located at position ( 1,3 ). If it moves in a velocity field
$$\mathbf{F}(x, y)=\left\langle x y-2, y^2-10\right\rangle$$
find its approximate location at time $t=1.05$.
The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines.(a) Use a sketch of the vector field $\mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j}$ to draw some flow lines. From your sketches, can you guess the equations of the flow lines?(b) If parametric equations of a flow line are $x=x(t)$, $y=y(t)$, explain why these functions satisfy the differential equations $d x / d t=x$ and $d y / d t=-y$. Then solve the differential equations to find an equation of the flow line that passes through the point $(1,1)$.
The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines.(a) Sketch the vector field $\mathbf{F}(x, y)=\mathbf{i}+x \mathbf{j}$ and then sketch some flow lines. What shape do these flow lines appear to have?(b) If parametric equations of the flow lines are $x=x(t)$, $y=y(t)$, what differential equations do these functions satisfy? Deduce that $d y / d x=x$.(c) If a particle starts at the origin in the velocity field given by $\mathbf{F}$, find an equation of the path it follows.