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# Vector calculus: review

These some are exercises for Vector calculus: course.

## First quarter

1. Give the number $t$ that makes $X=(3,2,1)$ and $Y=(2,t,t)$ perpendicular.
2. At time $t$ with $0 \leq t$, an object is at the position $$P[t]=(t^2 +1,\frac{1}{t+1},e^{2t}).$$ Calculate its velocity, v[t], and its acceleration, $a[t]$, as functions of $t$.
3. Calculate $X \times Y$ for $X=(1,-2,3)$ and $Y=(3,2,1)$.
4. Here are $xyz$-equations for two planes: $x+y-z=0$ and $x-y+z=0$. Explain how you can tell that these planes cut each other NOT at right angles.
5. A plane has an $xyz$-equation $x+y=2$. Give a vector perpendicular to the plane.
6. In an effort to find the line in which the planes $2x -y- z=2$ and $-4x+2y+2z=1$ intersect, a student multiplied the first one by $2$ and then added the result to the second. He got $0=5$. Explain the result.
7. Parametrically describe the line segment with endpoints $(-1,-1,-1)$ and $(1,1,1).$
8. "If $F$ is continuous at $x_{0}$ then $\lim_{x\rightarrow x_{0}}\frac{F(x)}{e^{x}}$ exists."' True or false? Explain.

## Second quarter

1. 1. Find a map $F:\mathbf{R}\rightarrow \mathbf{R}^{2}$ such that $F^{\prime}(t)=(e^{t},\sin t)$ and $F(0)=(0,1).$
2. Show that for any map a map $F:\mathbf{R}\rightarrow \mathbf{R}^{n},$ $\lim_{t\rightarrow t_{0}}\frac{1}{t}F(t)=0$ implies $\lim_{t\rightarrow t_0}F(t)=0.$
3. Give an example of a set $S$ and a point $p\in S$ such that $p$ is a limit point of $S$ and but not an interior point. Explain.
4. Find all partial derivatives of the following function: $f(x,y,z)=e^{xyz}+x\sin (yz).$
5. Find the point on the surface $z=x^{2}-y^{2}$ nearest to the origin. Explain.
6. Suppose $f,g:\mathbf{R}\rightarrow \mathbf{R}$ are continiuous functions. Let the map $H:\mathbf{R}\rightarrow \mathbf{R}^{2}$ be given by $H(x)=(f(x),g(x)).$ Prove that $H$ is continuous.

## First half

1. True or false?
1. The intersection of two linear subspaces is a linear subspace.
2. The empty set is a linear subspace.
3. The product of two linear functions is a linear function.
4. The acceleration is always perpendicular to the velocity.
5. There is only one natural parametrization of a straight line.
2. Give an example of:
1. a curve with curvature equal to 0,
2. a parametric curve that traces a circle 3 times clockwise,
3. a linear subspace in ${\bf R}^3$ with no vectors perpendicular to it,
4. a system of 2 linear equations with no solutions,
5. a system of 2 linear equations with exactly 2 solutions.
3. (a) Give the definition of a basis of a linear space. (b) Show that the vectors $(1,0,0), (1,1,0), (1,1,1)$ form a basis of ${\bf R}^3$.

## Third quarter

1. Let $f:\mathbf{R}\rightarrow \mathbf{R}$ be a functions such that $\int_{a}^{b}f(x)dx=1.$ Let $B=\{(x,y):a\leq x\leq b,c\leq y\leq d\}$ be a rectangle. Find $\int \int_{B}fdA.$
2. Find the volume of the region bounded by the surface $z=1-x^{2},$ the $xy$-plane and the planes $y=0$ and $y=1.$
3. Using the $\varepsilon-\delta$ definition, prove that the composition of two continuous functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}^{m}$ and $g:\mathbf{R}^{m}\rightarrow\mathbf{R}^{p}$ is continuous.
4. Let $F$ be a differentiable parametric curve. If $F^{\prime}(t)$ is perpendicular to $F(t)$ for all $t,$ show that $||F(t)||$ is constant.

## Second half

1. Given vector field $F[x,y]=(x,\frac{1}{2}y)$. Choose 10-15 points on the plane and pencil in the field vector $F[x,y]$ with tail at $(x,y)$. There are a few families of trajectories in this vector field. Pencil in a few trajectories of each type.
2. Suppose that a mass $M$ is fixed at the origin in space. When a particle of unit mass is placed at the point $(x,y)$ other than the origin, it is subjected to a force $G[x,y]$ of gravitational attraction. Plot the vector field $G[x,y]$, if the magnitude (length) of $G[x,y]$ is $\frac{kM}{r^2}$, where $r=\sqrt{x^2+y^2}$.
3. Here is a plot of a few trajectories (and vectors) of a vector field. On the basis of the plot, determine if it is a gradient field or not. Explain.

1. Use Gauss's formula to evaluate the flow of the vector field $F[x,y,z]=(z+y,x+y,z)$ across the surface of the 3D box with a slanted top consisting of all points $(x,y,z)$ with $0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq x+1$.
2. Find the area of that part of the plane $z=1+2x+2y$ that lies directly above the region in the $xy$-plane bounded by the parabolas $y=x^2-1$ and $y=-x^2+1.$

1. Find the divergence and the rotation of the vector field $F[x,y]=(x^2y,xy\sin y)$.
2. Measure the flow of the vector field $G[x,y]=(e^{x}+y,e^{y})$ ALONG the boundary of the rectangle with corners at $(0,0),(1,0),(0,1),(1,1)$.
3. Suppose a closed curve is located within the unit circle. Is the flow of $H[x,y]=(\frac{x^3}{3}-2x,\frac{y^3}{3})$ ACROSS this curve negative or positive? Explain.

1. Find a representation of a straight line $y=ax$ in polar coordinates.
2. You are faced with a hand calculation of $$\displaystyle\int\int_{R}f(x)dxdy,$$where $R$ is the two-dimensional region consisting of everything bounded by the curves $y=x^2,y=x^2+1$ and the lines $x=0,x=1$. Switch to new, convenient for integration, coordinates $u,v$ by indicating what $u,v$ are in terms of $x,y$ and describe $R$ in terms of $u,v$.
3. By hand calculation, evaluate $$\displaystyle\int\int_{R}1dxdy,$$ where $R$ is given in $u,v$ coordinates as $0 \leq u \leq 1,1 \leq v \leq 3$, and $x=u^2,y=v+u^2$.

1. You are faced with a hand calculation of $\displaystyle\int\int\int_{R}f(x,y,z)\,dxdydz$, where $R$ is the 3D region bounded from above by the unit sphere and from below by the xy-plane. Describe $R$ in a way convenient for integration.
2. You are faced with a hand calculation of $\displaystyle\int\int\int_{R}f(x,y,z)\,dxdydz$, where $R$ is the 3D region consisting of everything bounded by the planes $y=x,y=x+1,y=-x+1,y=-x+2,z=0,z=2$. Switch to new, convenient for integration, coordinates $u,v,w$ by indicating what $u,v,w$ are in terms of $x,y,z$ and describe $R$ in terms of $u,v,w$.
3. Find the volume conversion factor $V_{(u,v,w)}[x,y,z]$ of the transformation $x[u,v,w]=u^2v,y[u,v,w]=v^2,z[u,v,w]=w^2e^{u}$.

1. You are faced with a hand calculation of $\displaystyle\int\int_{R}f(x,y)dxdy$, where $R$ is the two-dimensional region consisting of everything bounded by the curves $y=x^2,y=x^2+2$ and the lines $x+y=1,x+y=2$. (a) Sketch $R$ in $xy$-plane. (b) Switch to new, convenient for integration, coordinates $u,v$ by indicating what $u,v$ are in terms of $x,y$. (c) Sketch $R$ in $uv$-plane.
2. By hand calculation, evaluate $$\displaystyle\int\int_{R}2dxdy,$$ where $R$ is given in $u,v$ coordinates as $1 \leq u \leq 2,0 \leq v \leq 1$, and $x=u^2+2v,y=ue^{v}$.
3. You are faced with a hand calculation of $\displaystyle\int\int\int_{R}f(x,y,z)dxdydz$, where $R$ is the "ice-cream cone", i.e. the 3D region obtained by intersecting the cone $z^2=x^2+y^2,z \geq 0$, and the sphere $x^2+y^2+z^2=2$. Describe $R$ in a way convenient for integration.
4. Find the volume conversion factor $V_{(x,y,z)}[u,v,w]$ of the transformation $x[u,v,w]=u^2+v+1,y[u,v,w]=v^2+w+2,z[u,v,w]=w^2+u+3$.
5. Use Gauss's formula to evaluate the flow of the vector field $F[x,y,z]={z^2,y,x^2}$ across the surface of the pyramid bounded by the coordinate planes and the first octant part of the plane with equation $x+y+z=1$.
6. Find by integration the area of that part of the plane $2x+3y+z=6$ that lies in the first octant.