Session 2
Lectures 5 - 8
- 12.6 Cylinders and Quadric Surfaces
- 13.1 Vector Functions and Space Curves
- 13.2 Derivatives and Integrals of Vector Functions
- 13.3 Arc Length and Curvature
- 13.4 Motion in Space: Velocity and Acceleration
12.6 Cylinders and Quadric Surfaces
A cylinder is a surface that consists of all lines (rulings) parallel to a given line and passing through a given plane curve.
- If one variable is missing from an equation of a surface, then the surface is a cylinder with rulings parallel to that variable’s axis
- A quadric surface is the graph of a second-degree equation in \(x\), \(y\), \(z\)
- To identify a quadric surface, find its traces (intersections with planes parallel to the coordinate planes)
12.6 Cylinders and Quadric Surfaces
cont.
The Standard Quadric Surfaces
- Ellipsoid: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)
- Elliptic paraboloid: \(\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}\)
- Hyperbolic paraboloid \(\frac{z}{c} = \frac{x^2}{a^2} - \frac{y^2}{b^2}\)
- Cone: \(\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}\)
- Hyperboloid of one sheet: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\)
- Hyperboloid of two sheets: \(-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)
Exercises
12.6 Cylinders and Quadric Surfaces
- What surface in \(\mathbb{R}^3\) is represented by the equation \(z = x^2\)?
- Identify the surface given by \(y = x^2 - z^2\) by finding its horizontal and vertical traces (with respect to y)
13.1 Vector Functions and Space Curves
A vector function \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\) assigns a vector to each value of the parameter \(t\)
- A vector function is a function whose domain is a set of real numbers and whose range is a set of vectors.
- \(\displaystyle\lim_{t \to a} \mathbf{r}(t) = \left\langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \right\rangle\), provided the limits of the component functions exist
- \(\mathbf{r}\) is continuous at \(a\) if \(\displaystyle\lim_{t \to a} \mathbf{r}(t) = \mathbf{r}(a)\)
- A space curve \(C\) is the set of all points \((f(t), g(t), h(t))\) for \(t \in I\) for some interval \(I\).
13.2 Derivatives and Integrals of Vector Functions
The derivative of a vector function is given componentwise \(\mathbf{r}'(t) = \lim_{h \to 0} \frac{\mathbf{r}(t+h) - \mathbf{r}(t)}{h} = \langle f'(t), g'(t), h'(t) \rangle\)
- \(\mathbf{r}'(t)\) is the tangent vector to the curve at the point defined by \(\mathbf{r}(t)\), provided \(\mathbf{r}'(t)\) exists and \(\mathbf{r}'(t) \neq \mathbf{0}\)
- Integration is also componentwise: \(\int \mathbf{r}(t)\, dt = \left\langle \int f(t)\,dt, \int g(t)\,dt, \int h(t)\,dt \right\rangle\)
13.2 Derivatives and Integrals of Vector Functions
cont.
Differentiation Rules
Let \(\mathbf{u}\), \(\mathbf{v}\) be differentiable vector functions, \(c\) a scalar, and \(f\) a scalar function.
- \(\frac{d}{dt}[\mathbf{u}(t) + \mathbf{v}(t)] = \mathbf{u}'(t) + \mathbf{v}'(t)\)
- \(\frac{d}{dt}[c\mathbf{u}(t)] = c\mathbf{u}'(t)\)
- \(\frac{d}{dt}[f(t)\mathbf{u}(t)] = f'(t)\mathbf{u}(t) + f(t)\mathbf{u}'(t)\)
- \(\frac{d}{dt}[\mathbf{u}(t) \cdot \mathbf{v}(t)] = \mathbf{u}'(t) \cdot \mathbf{v}(t) + \mathbf{u}(t) \cdot \mathbf{v}'(t)\)
- \(\frac{d}{dt}[\mathbf{u}(t) \times \mathbf{v}(t)] = \mathbf{u}'(t) \times \mathbf{v}(t) + \mathbf{u}(t) \times \mathbf{v}'(t)\)
Exercises
13.1 Vector Functions and Space Curves
13.2 Derivatives and Integrals of Vector Functions
The trajectory of a particle is given by \(\mathbf{r}(t) = \langle \sqrt{3}t^2, 2t^3, \sqrt{6}t^2 \rangle\). Let \(C\) denote the corresponding space curve.
- Find an equation for the tangent line to \(C\) at the point \((4\sqrt{3}, 16, 4\sqrt{6})\)
13.3 Arc Length and Curvature
- The length of a space curve \(\mathbf{r}(t)\), \(t \in [a,b]\) is given by: \[
L = \int_a^b |\mathbf{r}'(t)|\, dt = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2 + [h'(t)]^2}\, dt
\]
- The curvature of a curve is defined by \(\kappa(t) = \frac{|\mathbf{T}'(t)|}{|\mathbf{r}'(t)|}\) or (more usefully): \[
\kappa(t) = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}
\]
Exercises
13.3 Arc Length and Curvature
- Recall the last exercise. Now find the length of \(C\) on the interval \(t \in [0,\sqrt{8}]\).
13.4 Motion in Space: Velocity and Acceleration
If a particle’s position at time \(t\) is \(\mathbf{r}(t)\), then
- Velocity: \(\mathbf{v}(t) = \mathbf{r}'(t)\)
- Speed: \(|\mathbf{v}(t)| = |\mathbf{r}'(t)|\)
- Acceleration: \(\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t)\)
Exercises
13.4 Motion in Space: Velocity and Acceleration
In this problem all coordinates are measured in meters and time is measured in seconds. At time \(t = 0\) a ladybug, named Sam, is at position \((1, 1, 1)\) and is flying with constant velocity \(\langle 1, 2, 3 \rangle\) meters per second. A sensor placed at \((3, 6, 7)\) can detect ladybug motion that occurs within a sphere of radius \(7\) meters. Does the sensor detect Sam? If so, at what time is Sam last detected by the sensor?
Fin
This concludes session 2.