Vector Functions. Curves

Construct the hodograph of the vector function $(-\infty < t < +\infty )$:

Prove that the hodograph of the vector function

lies on a sphere.

Prove that the hodograph of the vector function

lies in some plane, and find the equation of this plane.

Prove that if $\lim_{t \rightarrow t_{0}} \mathbf{r}(t) = \mathbf{a}$, then $\lim_{t \rightarrow t_{0}}\left|\mathbf{r}(t)\right| = \left|\mathbf{a}\right|$. Is the converse statement true?

Prove that the vector function $\mathbf{r}(t) = t \mathbf{i}+\sin t \mathbf{j}$ is infinitesimal as $t \rightarrow 0$.

Find the limit of the vector function:

Prove that for the vector function $\mathbf{r}(t)$ to have, as $t \rightarrow t_{0}$, a limit equal to $\mathbf{a}$, it is necessary and sufficient that it can be represented in the form

where $\alpha (t)$ is a vector function infinitesimal as $t \rightarrow t_{0}$.

Prove that if $\lim_{t \rightarrow t_{0}} \mathbf{r}(t) = \mathbf{a}$ and $\lim_{t \rightarrow t_{0}} f(t) = \lambda$, then

Prove that if $\lim_{t \rightarrow t_{0}} r_{1}(t) = \mathbf{a}, \lim_{t \rightarrow t_{0}} \mathbf{r}_{2}(t) = \mathbf{b}$, and $\lim_{t \rightarrow t_{0}} \mathbf{r}_{3}(t) = \mathbf{c}$, then:

Prove that if the scalar functions $\lambda_{1}(t), \lambda_{2}(t)$ and the vector functions $\mathbf{r}_{1}(t), \mathbf{r}_{2}(t), \mathbf{r}_{3}(t)$ are continuous at the point $t_{0}$, then the following functions are also continuous at this point:

Find the derivative of the vector function $\mathbf{r}(t)$ and write the equation of the tangent at an arbitrary point of its hodograph, if:

Find the derivative of the functions:

Prove that if the length of the vectors $\mathbf{r}(t)$ is constant in a neighborhood of the point $t_{0}$ and the derivative $\mathbf{r}^{\prime }\left(t_{0}\right)$ exists, then the vectors $\mathbf{r}\left(t_{0}\right)$ and $\mathbf{r}^{\prime }\left(t_{0}\right)$ are orthogonal. What is the mechanical meaning of this fact?

Prove that for the vectors $\mathbf{r}(t)$ and $\mathbf{r}^{\prime }(t)$ to be orthogonal at every point of some interval, it is necessary and sufficient that the scalar function $\left|\mathbf{r}(t)\right|$ be constant on this interval.

Prove that for a differentiable vector function $\mathbf{r}(t)$, nonvanishing on the interval $(a ; b)$, to have constant direction (i.e., for the vector $\mathbf{r}(t)$ to be collinear, for any $t \in (a ; b)$, with, for example, the vector $\mathbf{r}((a+b) / 2)$), it is necessary and sufficient that the vectors $\mathbf{r}(t)$ and $\mathbf{r}^{\prime }(t)$ be collinear.

Let the vector function $\mathbf{r} = \mathbf{r}(t)$ have a derivative at the point $t_{0}$. Will the function $\left|\mathbf{r}(t)\right|$ be differentiable at this point? Are the equalities $\left|\mathbf{r}^{\prime }\right| = |\mathbf{r}|^{\prime }$ and $\left(\mathbf{r}, \mathbf{r}^{\prime }\right) = \left|\mathbf{r}\right||\mathbf{r}|^{\prime }$ valid at this point?

Prove that if

where $x_{1}, x_{2}, x_{3}$ — continuously differentiable functions of their arguments, and $x, y, z$ are continuously differentiable functions of $t$, then

Construct the hodographs of the vector function

and its derivative.

Using the definition of the derivative of a vector function, prove the formulas:

Prove that if $\mathbf{r} = \mathbf{a} \cos \omega t+\mathbf{b} \sin \omega t$, where $\omega$, $\mathbf{a}$ and $\mathbf{b}$ are constants, then:

Prove that if $\mathbf{r} = \mathbf{a} e^{\omega t}+\mathbf{b} e^{-\omega t}$, where $\omega$, $\mathbf{a}$, and $\mathbf{b}$ are constants, then

Prove that if the vector function $\mathbf{r}(t)$ is differentiable at the point $t_{0}$, then it is also continuous at it.

Let the vector function $\mathbf{r}(t)$ satisfy the differential equation $\mathbf{r}^{\prime \prime } = \left[\mathbf{r}^{\prime }, \mathbf{a}\right]$, where $\mathbf{a}$ is a constant vector. Express in terms of $\mathbf{a}$ and $\mathbf{r}^{\prime }$:

Let, for a vector function $\mathbf{r}(t)$ twice differentiable on the interval $[a ; b]$, the conditions

hold at all points of this interval. Prove that then the hodograph of the vector function $\mathbf{r}(t)$ lies in some plane.

Prove that if, for a vector function $r(t)$ twice differentiable on the interval $[a ; b]$, the vectors $\mathbf{r}^{\prime }(t)$ and $\mathbf{r}^{\prime \prime }(t)$ are nonzero and collinear at all points of this interval, then the hodograph of the vector function $\mathbf{r}(t)$ is a line segment.

Prove that the hodograph of the vector function $\mathbf{r}(t) = \mathbf{a}+t \mathbf{b}+t^{2} \mathbf{c}$, $t \in \mathbb {R}$, where $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$ are constant vectors, with the vectors $\mathbf{b}$ and $\mathbf{c}$ not collinear, is a parabola. What will the hodograph be if the vectors $\mathbf{b}$ and $\mathbf{c}$ are collinear?

Prove that the hodograph of the vector function $\mathbf{r}(t) = \mathbf{a}+\cos t \mathbf{b} + \sin t \mathbf{c}, 0 \leqslant t \leqslant 2 \pi$, where $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$ are constant vectors, with the vectors $\mathbf{b}$ and $\mathbf{c}$ not collinear, is an ellipse.

Prove that the trajectory of a material point moving under the action of a central force is planar.

The trajectory of motion of a point is given in cylindrical coordinates:

where $t$ is the time, $\rho (t), \varphi (t), z(t)$ are known functions. Find:

The trajectory of a moving point is given in spherical coordinates:

where $t$ is the time, $\rho (t), \varphi (t), \theta (t)$ are known functions. Find the magnitude of the instantaneous velocity.

Let $m$ be the mass of a point, $\mathbf{F} = \mathbf{F}(t)$ the force acting on it, $\mathbf{r} = \mathbf{r}(t)$ the law of motion of the point, $\mathbf{a} = \mathbf{a}(t)$ its acceleration, $W = W(t)$ the kinetic energy ($t$ is the time, $m$ a constant). From Newton's law $\mathbf{F} = m \mathbf{a}$, derive the formula $d W = (\mathbf{F}, d \mathbf{r})$.

Under the conditions of the previous problem, prove the formula $d \mathbf{N} = \mathbf{M} d t$, where $\mathbf{N}$ is the moment of momentum of the point relative to an arbitrarily chosen origin $O$, and $\mathbf{M}$ is the moment of the force $\mathbf{F}$ relative to the point $O$. (The momentum of a material point is the vector equal to the product of its mass and velocity. If some vector $\mathbf{b}$ is applied at the point $P$, then the moment of the vector $\mathbf{b}$ relative to the point $O$ is called the vector $[\overline{OP}, \mathbf{b}]$.)

Give an example of a differentiable vector function $\mathbf{r}(t)$, $a \leqslant t \leqslant b$, for which there is no point $\xi \in [a ; b]$ such that $\mathbf{r}(b) -\mathbf{r}(a) = \mathbf{r}^{\prime }(\xi )(b-a)$ (i.e., show that for vector functions, the analogue of the Lagrange finite-increment formula does not hold in this sense).

Prove that for a vector function $\mathbf{r}(t)$, differentiable on the interval $(a ; b)$, to be constant (i.e., $\mathbf{r}(t) = \mathbf{c}, a < t < b$, $\mathbf{c}$ a constant vector), it is necessary and sufficient that the derivative $\mathbf{r}^{\prime }(t)$ be identically zero on the interval $(a ; b)$.

Write the parametric equation of the involute of a circle, i.e., the trajectory of the end of a taut thread unwinding from a fixed round flat spool.

The line $OL$, not perpendicular to the axis $Oz$, rotates uniformly about it with constant angular velocity $\omega$. A point $M$ moves along the line $OL$:

Prove that the equations

and

are parametrizations of one and the same curve.

Prove that the equations

and

are parametrizations of one and the same curve. How does the point move along this curve as the parameter $t$ increases from $-\infty$ to $+\infty$?

Show that the curve $x = e^{a t} \cos t, y = e^{a t} \sin t, \quad z = e^{a t}$ lies on the cone $z^{2} = x^{2}+y^{2}$.

Show that the curve

$x = t /\left(1+t^{2}+t^{4}\right), \quad y = t^{2} /\left(1+t^{2}+t^{4}\right), \quad z = t^{3} /\left(1+t^{2}+t^{4}\right)$ is a spherical curve.

Under what condition on the matrix $\left(a_{i \jmath }\right)(i, j = 1,2,3)$ is the curve

a planar curve?

Prove that the projection of Viviani's curve

onto the plane of the variables $x$ and $z$ is an arc of a parabola.

Find the projection of the curve $x = e^{t} \sin t, y = e^{t} \cos t, z = t,-\infty < t < +\infty$, onto the plane of the variables $x$ and $y$.

Prove that the projection of the helix

onto the plane of the variables $y$ and $z$ is a sine curve.

Prove that under a translation of the origin to the point $O_{1} = (0 ; 0 ; \beta b)$ and a rotation of the $x$- and $y$-axes about the new $z$-axis through an angle $\beta$, the representation of the helix $x = a \cos t, y = a \sin t$, $z = b t$ can be brought to the form $x_{1} = a \cos t_{1}, y_{1} = a \sin t_{1}, z_{1} = b t_{1}$. This shows that the helix can slide along itself.

Find the equation of the tangent to the curve:

Write the equation of the tangent to the curve

at $t = \pi / 2$. What angle does this tangent make with the axis $Oz$?

Find the equation of the tangent line and the normal plane of the curve $x = t^{4}, y = t^{3}, z = t^{2}$ at an arbitrary point of it.

Find the tangent to Viviani's curve (see problem 42) parallel to the plane $y = 0$.

At which points is the tangent to the curve $x = 3 t-t^{3}, y = 3 t^{2}, z = 3 t+t^{3}$ parallel to the plane $3 x+y+z+2 = 0$?

Find the normal plane of the curve $z = x^{2}+y^{2}, y = x$, perpendicular to the line $x = y = z$.

Find the tangent to the curve $x^{2}+y^{2} = 10, y^{2}+z^{2} = 25$ at the point $(1 ; 3 ; 4)$.

Find the cosines of the angles with the coordinate axes made by the tangents to the curve $x^{2} = 2 a z, y^{2} = 2 b z$.

A tangent is drawn to the curve $y^{2} = 2 p x, z^{2} = 2 q x$ at the point where $x = (p+q) / 2$. Find the length of the segment of this tangent from the point of tangency to the plane $x = 0$.

Prove that the normal planes of the curve $x = a \cos t, y = a \sin \alpha \sin t, z = a \cos \alpha \sin t$ pass through the line $x = 0, z + y \operatorname {tg} \alpha = 0$.

Prove that the tangents to the curve $x^{2} = 3 y, 2 x y = 9 z$ make a constant angle with some fixed direction.

The coordinates of the points of a certain curve satisfy the relation

Prove that the tangents to this curve are tangent to the sphere $x^{2}+y^{2} + z^{2} = a^{2}$.

Prove that the tangents to the curve

intersect the plane of the variables $x$ and $y$ in the circle $x^{2}+y^{2} = 4 a^{2}$.

Write the equation of the tangent and the normal plane of the curve $x = t, y = t^{2}, z = t^{3}$ at the point $(1 ; 1 ; 1)$. What curve results from the intersection of the tangents with the plane of the variables $x, y$?

Find the equation of the normal plane at an arbitrary point of the curve $x^{2}+y^{2} = 1, y^{2}+z^{2} = 1(y \neq \pm 1)$.

Prove that all normal planes of Viviani's curve $x = a \sin^{2} t, y = a \sin t \cos t, z = a \cos t$ pass through the origin.

Prove that if all normal planes of a space curve pass through a fixed point, then the curve is spherical.

Prove that the curve $x = e^{t} \cos t, y = e^{t} \sin t, z = e^{t}$ intersects all the generators of the cone $x^{2}+y^{2} = z^{2}$ at one and the same angle.

Prove that the curves of intersection of the cylinders $y^{2}+z^{2} = b^{2}$ with the surface $xy = az$ intersect, at a right angle, all the generators of this surface belonging to one family.

Prove that the curve $x = a \operatorname {tg} t, y = b \cos t, z = b \sin t$ lies on the surface of a paraboloid and intersects, at a right angle, all its generators of one family.

The curve called a loxodrome is defined by the equation

where $\theta$ is the latitude and $\varphi$ the longitude of a point on the sphere. Prove that it intersects the meridians of the sphere at an angle $\alpha$ whose tangent equals $a$.

Fig. 24.8

Let us formulate the definition of the stereographic projection of a plane onto a sphere tangent to it. Suppose that in the space $\mathbb {R}^{3}$ a Cartesian rectangular coordinate system $x, y, z$ is fixed, and a sphere of radius $a / 2$ with center at the point $B = (0 ; 0 ; a / 2)$, tangent to the plane of the variables $x$ and $y$ (Fig. 24.8), is given. Its equation has the form

Let us join by a line the top point of the sphere, i.e., the point $A = (0 ; 0 ; a)$, with an arbitrarily fixed point $M = (x ; y ; 0)$ of the plane of the variables $x$ and $y$. Then the point $M_{1} = \left(x_{1} ; y_{1} ; z_{1}\right)$ at which this line intersects the sphere (53) is called the stereographic projection of the point $M$ onto the sphere (53), and the point $M$ is called the stereographic projection of the point $M_{1}$ of the sphere onto the plane in question. The stereographic projection establishes a one-to-one correspondence between the points of the plane and the sphere with the point $A$ removed. The coordinates of the points $M$ and $M_{1}$ are related by

Prove that the stereographic projection gives a conformal map, i.e., that curves in the plane intersect at the same angle as their images on the sphere.

Prove that the curve $\rho = e^{\lambda \varphi }$, situated in the plane of the variables $x$ and $y$ so that the polar axis coincides with the positive part of the $x$-axis, maps to a loxodrome under the stereographic projection (54).

Prove that circles on the sphere, under the stereographic projection, go over into circles or lines in the plane.

The central projection of the coordinate plane of the variables $x, y$ onto the hemisphere $x^{2}+y^{2}+z^{2} = 2 a z, 0 \leqslant z < a$, consists of the following. An arbitrarily fixed point $M = (x ; y ; 0)$ of the plane of the variables $x$ and $y$ is joined by a line to the center of the given hemisphere, i.e., to the point $A = (0 ; 0 ; a)$. The point $M_{1}$ at which this line intersects the hemisphere is taken as the image of the point $M$ on the hemisphere.

Prove that this projection is not a conformal map, i.e., it does not, in general, preserve the angles between curves.

Find the derivative of the arc length with respect to the parameter for:

Let a plane curve $\Gamma$ be given in polar coordinates by the equation $\rho = \rho (\varphi )$, where the function $\rho (\varphi )$ is continuously differentiable on some interval $[a ; b]$. Prove that if $s = s(\varphi )$ is the arc length of the curve $\Gamma$, measured from its starting point, then

Find the derivative of the arc length for the following curve given in polar coordinates:

Prove that the boundary of a bounded convex figure in the plane is a rectifiable curve.

Prove that when curves are joined their lengths add: if $\Gamma = \Gamma_{1} \cup \Gamma_{2}$, then $S_{\Gamma } = S_{\Gamma_{1}}+S_{\Gamma_{2}}$.

Find the curvature and the radius of curvature at an arbitrary point:

Find the curvature and the center of curvature at an arbitrary point:

Find the curvature of the curve at an arbitrary point:

Find the evolute of the curve:

Find the curvature at the point $M_{0}$ of the graph of the function $y = y(x)$ given implicitly by the equation $F(x, y) = 0$ :

Find the greatest curvature of the curve:

Prove that the radius of curvature of the parabola $x^{2} = 2 p y$ equals $R = p / \cos^{3} \alpha$, where $\alpha$ is the angle of inclination of the tangent to the axis of abscissas.