### Key Concepts

### 3.1 Vector-Valued Functions and Space Curves

- A vector-valued function is a function of the form $\text{r}(t)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}$ or $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k},$ where the component functions
*f, g,*and*h*are real-valued functions of the parameter*t*. - The graph of a vector-valued function of the form $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}$ is called a
*plane curve*. The graph of a vector-valued function of the form $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k}$ is called a*space curve*. - It is possible to represent an arbitrary plane curve by a vector-valued function.
- To calculate the limit of a vector-valued function, calculate the limits of the component functions separately.

### 3.2 Calculus of Vector-Valued Functions

- To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function.
- Many of the properties of differentiation from the Introduction to Derivatives also apply to vector-valued functions.
- The derivative of a vector-valued function $\text{r}\left(t\right)$ is also a tangent vector to the curve. The unit tangent vector $\text{T}\left(t\right)$ is calculated by dividing the derivative of a vector-valued function by its magnitude.
- The antiderivative of a vector-valued function is found by finding the antiderivatives of the component functions, then putting them back together in a vector-valued function.
- The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function.

### 3.3 Arc Length and Curvature

- The arc-length function for a vector-valued function is calculated using the integral formula $s\left(t\right)={\displaystyle {\int}_{a}^{t}\Vert {r}^{\prime}\left(u\right)\Vert \phantom{\rule{0.2em}{0ex}}du}.$ This formula is valid in both two and three dimensions.
- The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature.
- There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius.
- The principal unit normal vector at
*t*is defined to be

$$\text{N}\left(t\right)=\frac{{T}^{\prime}\left(t\right)}{\Vert {T}^{\prime}\left(t\right)\Vert}.$$ - The binormal vector at
*t*is defined as $\text{B}\left(t\right)=\text{T}\left(t\right)\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{N}\left(t\right),$ where $\text{T}\left(t\right)$ is the unit tangent vector. - The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.
- The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point.

### 3.4 Motion in Space

- If $\text{r}\left(t\right)$ represents the position of an object at time
*t*, then $\text{r}\prime \left(t\right)$ represents the velocity and $\text{r\u2033}\left(t\right)$ represents the acceleration of the object at time*t.*The magnitude of the velocity vector is speed. - The acceleration vector always points toward the concave side of the curve defined by $\text{r}\left(t\right).$ The tangential and normal components of acceleration ${a}_{\text{T}}$ and ${a}_{\text{N}}$ are the projections of the acceleration vector onto the unit tangent and unit normal vectors to the curve.
- Kepler’s three laws of planetary motion describe the motion of objects in orbit around the Sun. His third law can be modified to describe motion of objects in orbit around other celestial objects as well.
- Newton was able to use his law of universal gravitation in conjunction with his second law of motion and calculus to prove Kepler’s three laws.