You CAN Ace Calculus | |
---|---|
17calculus > vector functions > equations |
Topics You Need To Understand For This Page
Calculus Main Topics
Single Variable Calculus |
---|
Multi-Variable Calculus |
Tools
math tools |
---|
general learning tools |
additional tools |
ATTENTION INSTRUCTORS: The new 2018 version of 17calculus will include changes to the practice problem numbering system. If you would like advance information to help you prepare for spring semester, send us an email at 2018info at 17calculus.com. |
Join Amazon Student - FREE Two-Day Shipping for College Students |
---|
How to Develop a Brilliant Memory Week by Week: 50 Proven Ways to Enhance Your Memory Skills |
Vector Functions Equations |
---|
This page contains the full list of vector function equations including links to the pages where they are discussed and derived. Which equation you use will depend on the data you have to work with. We start by giving the variables and their names. |
Variables | ||
---|---|---|
\(x(t),~y(t),~z(t)\) | ||
\(\vec{r}(t) = x(t)\vhat{i} + y(t)\vhat{j} + z(t)\vhat{k}\) | ||
vector function in terms of the arc length parameter, s |
\(\vec{r}(s) = x(s)\vhat{i} + y(s)\vhat{j} + z(s)\vhat{k}\) | |
Note - If the variable s is used on this page, it refers to the arc length parameter. | ||
Equations | ||
\(\vec{v}(t) = \vec{r}'(t)\) | ||
\(\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)\) | ||
\(\displaystyle{ \vhat{T}(t) = \frac{\vec{r}'(t)}{ \| \vec{r}'(t) \| } }\) | ||
\(\displaystyle{ \vhat{T}(t) = \frac{\vec{v}(t)}{ \| \vec{v}(t) \| } }\) | ||
\(\displaystyle{ \vhat{N}(t) = \frac{d\vhat{T}/dt}{ \| d\vhat{T}/dt \| } }\) | ||
\( \vec{a}(t) = a_{\vhat{T}}\vhat{T} + a_{\vhat{N}}\vhat{N}\) | ||
\(a_{\vhat{T}} = \vec{a} \cdot \vhat{T} \) | ||
\(\displaystyle{a_{\vhat{T}} = \frac{\vec{a} \cdot \vec{v}}{\|\vec{v}\|} }\) | ||
\(a_{\vhat{T}} = \| \vec{v} \|' \) | ||
\(a_{\vhat{N}} = \vec{a} \cdot \vhat{N} \) | ||
\(a_{\vhat{N}} = \|\vec{v}\| \|\vhat{T}'\|\) | ||
\(\displaystyle{a_{\vhat{N}} = \frac{\|\vec{v} \times \vec{a}\|}{\|\vec{v}\|} }\) | ||
\(a_{\vhat{N}} = \sqrt{\|\vec{a}\|^2 - a_{\vhat{T}}^2}\) |