Chapter 3

Previous: Chapter 2

Curves and Motion in Space

![[Pasted image 20240908232403.png]]
A particle moving in 3D space. It's position at \(t\) can be described by 3 parametric equations
\(x = f(t)\), \(y=g(t)\), and \(z=h(t)\)

Position: \(\textbf{r} = \pmatrix{f(t) \\ g(t) \\ h(t)}\)

Velocity: \(\textbf{r} \prime (t) = \frac{d}{dt} \textbf{r}(t) = \pmatrix{f \prime(t)\\ g \prime(t) \\ h \prime (t)}\)

Acceleration: \(\textbf{r} \prime \prime (t) = \frac{d^2}{dt^2} \textbf{r}(t) = \pmatrix{f \prime \prime(t)\\ g \prime \prime(t) \\ h \prime \prime (t)}\)

Tangent Lines

A vector equation of the tangent line to a curve \(\textbf{r} = \pmatrix{f(t) \\ g(t) \\ h(t)}\) at the point where \(t = t_0\) is
\(\textbf{r} = \pmatrix{f(t_0) \\ g(t_0) \\ h(t_{0)}} +s\pmatrix{f \prime(t_0)\\ g \prime(t_0) \\ h \prime (t_{0)}}, s \in \textbf{R}\)

Line segments

![[Pasted image 20240908233416.png]]
The line segment joining two distinct point \(\textit{A}(x_{1}, y_{1}, z_{1})\) and \(\textit{B}(x_2,y_2,z_{2})\) has parametric representations:
\(\textbf{r}(t) = (1-t)\pmatrix{x_1\\y_1\\z_{1}}+ t\pmatrix{x_{2}\\ y_{2} \\ z_{2}} = \pmatrix{(1-t)x_{1}+ tx_{2}\\ (1-t)y_{1}+ y_{2}\\(1-t)z_{1}+ tz _2}\)

Circles (In a 2D space)

\(\textbf{r}(t) = \pmatrix{\textit{r}cost \\\textit{r}sint}, 0 \leq t \leq 2\pi\)

Definite Integrals

![[Pasted image 20240909100709.png]]
Let \(\textbf{r} = \pmatrix{f(t) \\ g(t) \\ h(t)} = (f(t))i + g(t)j + h(t)k\)
\(\int\textbf{r}(t)dt = (\int f(t)dt)i + (\int g(t)dt)j + (\int h(t)dt)k\)

Definite integrals

\(\int_{a}^{b}\textbf{r}(t)dt = (\int_{a}^{b} f(t)dt)i + (\int_{a}^{b} g(t)dt)j + (\int_{a}^{b} h(t)dt)k \space dt\)

The Smooth curve is defined by the function
\(\textbf{r} = \pmatrix{f(t) \\ g(t) \\ h(t)}, a \leq t \leq b\)
Length L can be obtained with:
\(L = \int_{a}^{b} \sqrt{(f \prime (t))^{2}+(g \prime (t))^{2}+ (h \prime (t))^{2}}\space dt\)

Line integrals

![[Pasted image 20240909000424.png]]
height = \(f(x(t), y(t))\)
surface: \(z = f(x, y)\)
curve \(C\): \(x=x(t), y=y(t)\)

The line integral of \(C\) is given by

2D

\(\int_{C}f(x,y) \space ds = \int^{b}_{a}f(x(t),y(t)) \cdot \sqrt{(x\prime (t))^{2}+ y\prime(t)^{2}} \space dt\)

3D

\(\int^{b}_{a}f(x(t),y(t),z(t)) \cdot \sqrt{(x\prime (t))^{2}+ (y\prime(t))^{2} + (z\prime(t))^2} \space dt\)

Orientation of curves

Positive orientation corresponds to increasing values of \(t\)
Negative orientation corresponds to decreasing values of \(t\)

Parametric surfaces

\(\textbf{r}(u,v)=x(u,v)i + y(u,v)j + z(u,v)k\)
![[Pasted image 20240909001150.png]]
for each \(u,v\) in D \(\textbf{r}(u,v)\) represents a point in space. The points constitute a surface.

Let \(P\) be the point on the parametric surface
\(\textbf{r}(u,v)=x(u,v)i + y(u,v)j + z(u,v)k, (u,v) \in D\)
with position vector \(\textbf{r}(u_0,v_{0})= x_{0}i+ y_{0}j + z_{0}k\) and let \(\textbf{n} = (\textbf{r}_{u} \times \textbf{r}_v)(u_{0},v_{0})\) be the vector \(\textbf{r}_{u} \times \textbf{r}_v\) evaluated at \((u_{0},v_{0})\)

A vector equation of the tangent plane at \(P\) to the surface is given by
\(\textbf{r} \cdot \textbf{n} = \pmatrix{x_{0}\\ y_{0}\\ z_{0}} \cdot \textbf{n}\)
where \(\textbf{r} = \pmatrix{x \\ y \\ z}\) is the position vector of any point on the tangent plane.

Next: Chapter 4