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Chapter 7: Problem 8

Find an appropriate parametrization for the given piecewise-smooth curve in \(\mathrm{R}^{3}\). The intersection of the cylinder \(y^{2}+z^{2}=1\) and the plane \(z=x\)

Short Answer

Expert verified
The parametrization is \( r(t) = (\sin(t), \cos(t), \sin(t)) \) for \( 0 \leq t < 2\pi \).

Step by step solution

01

Understand the Intersection

The intersection we need to parametrize is formed by the cylinder defined by the equation \( y^2 + z^2 = 1 \) and the plane \( z = x \). This means that for every point on the curve, \( z = x \) holds true. As we traverse the cylinder, we must ensure that the plane's equation is satisfied.
02

Use Cylinder Equation for Parameter

The equation \( y^2 + z^2 = 1 \) suggests using trigonometric functions for \( y \) and \( z \). Let \( y = \cos(t) \) and \( z = \sin(t) \), where \( t \) is any real parameter such that \( 0 \leq t < 2\pi \), allowing \( y^2 + z^2 = \cos^2(t) + \sin^2(t) = 1 \).
03

Incorporate Plane Constraint

Since we have established \( z = \sin(t) \), and the plane equation dictates \( z = x \), set \( x = \sin(t) \). This respects the plane's condition of \( z = x \) throughout the curve. Thus, the parametrization of the curve becomes \( x = \sin(t) \), \( y = \cos(t) \), and \( z = \sin(t) \).
04

Parametrization Result

Our final parametrization for the curve is given by the vector function \( r(t) = (\sin(t), \cos(t), \sin(t)) \), where \( 0 \leq t < 2\pi \). This parameterization adheres to both the cylinder and plane conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylinder Equation
When you encounter the term "cylinder equation," think of a surface in three-dimensional space that is shaped like a long tube. In mathematical terms, a cylinder can be represented by an equation like \( y^2 + z^2 = 1 \).
This equation defines the set of points that are on the surface of a cylinder centered around the x-axis.
  • The equation \( y^2 + z^2 = 1 \) specifies that every point on this cylinder is a unit distance from the x-axis.
  • Such a form is specific to a circular cylinder, where cross-sections perpendicular to the cylinder's axis are circles.
  • If you trace the surface, moving around the x-axis, you'll see that the values of \( y \) and \( z \) stay on the circle defined by this equation for any given \( x \).
Understanding the cylinder equation is key when dealing with problems involving cylindrical surfaces in 3D geometry, as it helps set bounds and shape for any parametrizations.
Trigonometric Parametrization
Trigonometric parametrization is a powerful technique used to represent curves or surfaces using trigonometric functions. It is especially useful with circular or elliptical shapes.
  • Consider the unit circle in a 2D plane described by \( x^2 + y^2 = 1 \). Using trigonometric parametrization, we can set \( x = \cos(t) \) and \( y = \sin(t) \), where \( t \) is the parameter that traces the angle on the unit circle.
  • In the context of our exercise, we apply similar logic to the cylinder's equation \( y^2 + z^2 = 1 \).
  • We let \( y = \cos(t) \) and \( z = \sin(t) \), ensuring that \( y^2 + z^2 = \cos^2(t) + \sin^2(t) = 1 \) holds true due to the Pythagorean identity.
This approach simplifies computations significantly and provides a neat and continuous representation of curves using a parameter \( t \).
Intersection of Surfaces
The intersection of surfaces is a fascinating concept in geometry. It involves finding common points that lie on two or more surfaces. In our case, we consider the intersection of a cylinder and a plane.
  • The cylinder is defined by \( y^2 + z^2 = 1 \), and the plane is defined by \( z = x \).
  • The challenge is to find a set of points that satisfy both conditions simultaneously. These points define a special curve on the surfaces.
  • For the curve parameterization, we ensure each point satisfies \( z = x \) in addition to the cylinder equation.
By aligning the trigonometric functions from the cylinder with the plane equation (setting both \( z \) and \( x \) to \( \sin(t) \)), we accurately capture all intersections under this condition.
Vector Function
A vector function is a mathematical expression that describes a curve using a vector to represent position in space as a function of a parameter. It shows how position changes as the parameter varies.
  • In our example, the vector function is represented as \( r(t) = (\sin(t), \cos(t), \sin(t)) \).
  • This notation indicates the positions \( x = \sin(t) \), \( y = \cos(t) \), and \( z = \sin(t) \) for the curve formed at any value of \( t \) in the given range.
  • Vector functions are particularly useful because they provide a compact way to capture and manipulate positions and directions in space by simply varying the parameter \( t \).
Through parametrization with a vector function, you can visualize paths and shapes in three dimensions, enabling a deeper understanding of how complex surfaces and curves interact.

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Most popular questions from this chapter

Evaluate \(\int_{c} f d s,\) where \(f(x, y, z)=z\) and \(\mathbf{c}(t)=(t \cos t, t \sin t, t)\) for \(0 \leq t \leq t_{0}\)

If \(S\) is the upper hemisphere \(\left[(x, y, z) | x^{2}+y^{2}+\right.\) \(\left.z^{2}=1, z \geq 0\right\\}\) oriented by the normal pointing out of the sphere, compute \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) for parts (a) and (b). (a) \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}\) (b) \(\mathbf{F}(x, y, z)=y \mathbf{i}+x \mathbf{j}\) (c) For each of these vector fields, compute \(\iint_{S}(\nabla \times \mathbf{F}) \cdot d \mathbf{S}\) and \(\int_{C} \mathbf{F} \cdot d \mathbf{s},\) where \(C\) is the unit circle in the \(x y\) plane traversed in the counterclockwise direction (as viewed from the positive \(z\) axis). (Notice that \(C\) is the boundary of \(S\). The phenomenon illustrated here will be studied more thoroughly in the next chapter, using Stokes" theorem.)

(a) Show that the path integral of \(f(x, y)\) along a path given in polar coordinates by \(r=r(\theta)\) \(\theta_{1} \leq \theta \leq \theta_{2},\) is $$\int_{\theta_{1}}^{\theta_{2}} f(r \cos \theta, r \sin \theta) \sqrt{r^{2}+\left(\frac{d r}{d \theta}\right)^{2}} d \theta$$ (b) Compute the are length of the path \(r=1+\cos \theta, 0 \leq \theta \leq 2 \pi\)

Evaluate \(\int_{e} \mathbf{F} \cdot d \mathbf{s},\) where \(\mathbf{F}(x, y, z)=y \mathbf{i}+2 x \mathbf{j}+y \mathbf{k}\) and the path \(c\) is defined by \(c(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\) \(0 \leq t \leq 1\)

Find an appropriate parametrization for the given piecewise-smooth curve in \(\mathrm{R}^{3}\). The triangle formed by traveling from the point (1,2,3) to \((0,-2,1),\) to \((6,4,2),\) and back to (1,2,3)
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