17Calculus Vector Functions - Equations
This page contains a 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.
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Variables | ||
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\(x(t),~y(t),~z(t)\) | ||
\(\vec{r}(t) = x(t)\vhat{i} + y(t)\vhat{j} + z(t)\vhat{k}\) | ||
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 - When the variable s is used on this page, it refers to the arc length parameter. | ||
Equations | ||
\(\vec{v}(t) = \vec{r}'(t)\) | ||
\(\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)\) | ||
\(\displaystyle{ \vhat{T}(t) = \frac{\vec{r}'(t)}{ \| \vec{r}'(t) \| } }\) | ||
\(\displaystyle{ \vhat{T}(t) = \frac{\vec{v}(t)}{ \| \vec{v}(t) \| } }\) | ||
\(\displaystyle{ \vhat{N}(t) = \frac{d\vhat{T}/dt}{ \| d\vhat{T}/dt \| } }\) | ||
\( \vec{a}(t) = a_{\vhat{T}}\vhat{T} + a_{\vhat{N}}\vhat{N}\) | ||
\(a_{\vhat{T}} = \vec{a} \cdot \vhat{T} \) | ||
\(\displaystyle{a_{\vhat{T}} = \frac{\vec{a} \cdot \vec{v}}{\|\vec{v}\|} }\) | ||
\(a_{\vhat{T}} = \| \vec{v} \|' \) | ||
\(a_{\vhat{N}} = \vec{a} \cdot \vhat{N} \) | ||
\(a_{\vhat{N}} = \|\vec{v}\| \|\vhat{T}'\|\) | ||
\(\displaystyle{a_{\vhat{N}} = \frac{\|\vec{v} \times \vec{a}\|}{\|\vec{v}\|} }\) | ||
\(a_{\vhat{N}} = \sqrt{\|\vec{a}\|^2 - a_{\vhat{T}}^2}\) | ||
\(\displaystyle{ K(t) = \frac{1}{\|\vec{v}\|} \left\| \frac{d\vec{T}}{dt} \right\| = \frac{\| \vec{T}'(t) \|}{\|\vec{r}'(t)\|} }\) | ||
\(\displaystyle{ K=\frac{\|\vec{v} \times \vec{a} \|}{\|\vec{v}\|^3} }\) | ||
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Topics You Need To Understand For This Page
Trig Formulas
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
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Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
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Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
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Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
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Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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Topics Listed Alphabetically
Single Variable Calculus |
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Multi-Variable Calculus |
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Differential Equations |
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Precalculus |
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