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fixed links in dyn and sta
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@jcrayb
jcrayb committed Jan 18, 2025
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30 changes: 15 additions & 15 deletions src/pages/dyn/particle_kinematics.astro
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Expand Up @@ -278,7 +278,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
</table>
<p class="figureCaption mt-2">
Velocity and acceleration in the polar basis. Compare to
Figure <a href="#rkv-fa">#rkv-fa</a>. Observe that
Figure <a href="#rkv-fa-c">#rkv-fa</a>. Observe that
<InlineEquation equation="\\hat{e}_r,\\hat{e}_\\theta" /> are not related to the path
(not tangent, not in the direction of movement), but
rather are defined only by the position vector. Note also
Expand Down Expand Up @@ -456,12 +456,12 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<DisplayEquation title="Derivative of general vectors." id="rkr-ed" background="True" derivation="True" equation="\\begin{aligned}\\dot{\\vec{a}} = \\underbrace{\\dot{a}\\hat{a}}_{\\operatorname{Proj}(\\dot{\\vec{a}}, \\vec{a})} +\\underbrace{\\vec{\\omega} \\times\\vec{a}}_{\\operatorname{Comp}(\\dot{\\vec{a}}, \\vec{a})}\\end{aligned}">
<p>
Using the same approach as <a
href="rvc.html#rvc-em">#rvc-em</a> we write <InlineEquation equation="\\vec{a} =a\\hat{a}" /> and differentiate this and use <a
href="/dyn/vectors#rvc-em2">#rvc-em</a> we write <InlineEquation equation="\\vec{a} =a\\hat{a}" /> and differentiate this and use <a
href="#rkr-ew">rkr-ew</a> to find:

<DisplayEquation equation="\\begin{aligned}\\dot{\\vec{a}} &amp;= \\frac{d}{dt} \\big( a \\hat{a} \\big) \\\\&amp;= \\dot{a} \\hat{a} + a \\dot{\\hat{a}} \\\\&amp;= \\dot{a} \\hat{a} + a (\\vec\\omega \\times \\hat{a}) \\\\&amp;= \\dot{a} \\hat{a} + \\vec\\omega \\times (a \\hat{a}) \\\\&amp;= \\dot{a} \\hat{a} + \\vec\\omega \\times \\vec{a}.\\end{aligned}" />

Comparing this to <a href="rvc.html#rvc-em">#rvc-em</a>
Comparing this to <a href="/dyn/vectors#rvc-em2">#rvc-em</a>
shows that the two components are the projection and the
complementary projection, respectively.
</p>
Expand Down Expand Up @@ -617,7 +617,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<p>
Using <a href="#rkr-el">#rkr-el</a> and the scalar
triple product formula <a
href="rvi.html#rvi-es">#rvi-es</a> gives:
href="https://en.wikipedia.org/wiki/Triple_product">#rvi-es</a> gives:

<DisplayEquation equation="\\begin{aligned}\\vec{a} \\cdot \\dot{\\vec{a}}&amp;= \\vec{a} \\cdot \\big( \\vec{\\omega} \\times \\vec{a} \\big) \\\\&amp;= \\vec{\\omega} \\cdot \\big( \\vec{a} \\times \\vec{a} \\big) \\\\&amp;= 0.\\end{aligned}" />
</p>
Expand All @@ -627,7 +627,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<p>
We first consider the dot product <InlineEquation equation="\\vec{a} \\cdot \\vec{b}" /> and show that this is not changing with
time. We do this by using the scalar triple product
formula <a href="rvi.html#rvi-es">#rvi-es</a> to find:
formula <a href="https://en.wikipedia.org/wiki/Triple_product" target="_blank">#rvi-es</a> to find:

<DisplayEquation equation="\\begin{aligned}\\frac{d}{dt} \\big( \\vec{a} \\cdot \\vec{b} \\big)&amp;= \\dot{\\vec{a}} \\cdot \\vec{b} + \\vec{a} \\cdot \\dot{\\vec{b}} \\\\&amp;= (\\vec{\\omega} \\times \\vec{a}) \\cdot \\vec{b} + \\vec{a} \\cdot (\\vec{\\omega} \\times \\vec{b}) \\\\&amp;= \\vec{b} \\cdot (\\vec{\\omega} \\times \\vec{a}) + \\vec{b} \\cdot (\\vec{a} \\times \\vec{\\omega}) \\\\&amp;= \\vec{b} \\cdot (\\vec{\\omega} \\times \\vec{a}) - \\vec{b} \\cdot (\\vec{\\omega} \\times \\vec{a}) \\\\&amp;= 0.\\end{aligned}" />

Expand Down Expand Up @@ -722,7 +722,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
These equations are definitions of the basis vectors, so
the only thing to derive is the alternative formula for
<InlineEquation equation="\\hat{e}_n" />. Using the definition of <InlineEquation equation="\\hat{e}_t" /> above
and <a href="rvc.html#rvc-eu">#rvc-eu</a>, we see that
and <a href="/dyn/vectors#rvc-eu">#rvc-eu</a>, we see that

<DisplayEquation equation="\\dot{\\hat{e}}_t = \\dot{\\hat{v}} = \\frac{1}{v} \\operatorname{Comp}(\\dot{\\vec{v}}, \\vec{v}) = \\frac{1}{v} \\operatorname{Comp}(\\vec{a}, \\vec{v})." />

Expand Down Expand Up @@ -824,7 +824,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"

Now the basis vector derivatives are given by the cross
product by <InlineEquation equation="\\vec{\\omega}" /> from <a
href="rkr.html#rkr-ew">#rkr-ew</a>, so we can evaluate
href="#rkr-ew">#rkr-ew</a>, so we can evaluate
the expressions <a href="#rkt-ek">#rkt-ek</a> for
curvature and torsion to give:

Expand Down Expand Up @@ -857,7 +857,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<p>
We can use the expression <a href="#rkt-ew">#rkt-ew</a>
for <InlineEquation equation="\\vec{\\omega}" /> together with <a
href="rkr.html#rkr-ew">#rkr-ew</a> to find the basis
href="#rkr-ew">#rkr-ew</a> to find the basis
vector derivatives:

<DisplayEquation equation="\\begin{aligned}\\dot{\\hat{e}}_t &amp;= \\vec{\\omega} \\times \\hat{e}_t= (v\\tau \\,\\hat{e}_t + v \\kappa \\,\\hat{e}_b) \\times \\hat{e}_t= v \\kappa \\,\\hat{e}_n \\\\\\dot{\\hat{e}}_n &amp;= \\vec{\\omega} \\times \\hat{e}_n= (v\\tau \\,\\hat{e}_t + v \\kappa \\,\\hat{e}_b) \\times \\hat{e}_n= - v \\kappa \\,\\hat{e}_t + v \\tau \\,\\hat{e}_b \\\\\\dot{\\hat{e}}_b &amp;= \\vec{\\omega} \\times \\hat{e}_b= (v\\tau \\,\\hat{e}_t + v \\kappa \\,\\hat{e}_b) \\times \\hat{e}_b= -v \\tau \\,\\hat{e}_n.\\end{aligned}" />
Expand Down Expand Up @@ -945,7 +945,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
direction of <InlineEquation equation="\\vec{v}" />, that
<InlineEquation equation="\\operatorname{Comp}(\\vec{a},\\vec{v})" /> does not depend
on the magnitude of <InlineEquation equation="\\vec{v}" />, and equation <a
href="rvv.html#rvv-em">#rvv-em</a> for the magnitude of
href="#rvv-em">#rvv-em</a> for the magnitude of
the complementary projection.
</p>
</DisplayEquation>
Expand Down Expand Up @@ -1086,7 +1086,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<InlineEquation equation="\\hat{e}_b" />. At the current instant when the particles
are at the same position, this means that <InlineEquation equation="\\hat{e}_r = -\\hat{e}_n" /> and <InlineEquation equation="\\hat{e}_\\theta = \\hat{e}_t" />. Using the
circular motion expressions <a
href="rke.html#rke-ep">#rke-ec</a> we have that the
href="https://en.wikipedia.org/wiki/Circular_motion" target="_blank">#rke-ec</a> we have that the
velocity of \(Q\) is:

<DisplayEquation equation="\\begin{aligned} \\vec{v}_Q &amp;= v_Q \\,\\hat{e}_\\theta \\\\ &amp;= \\dot{s}_Q \\,\\hat{e}_\\theta \\\\ &amp;= \\dot{s}_P \\,\\hat{e}_t = \\vec{v}_P. \\\\ \\end{aligned}" />
Expand Down Expand Up @@ -1179,7 +1179,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<DisplayEquation equation="\\begin{aligned}\\|\\vec{r}'' \\times \\vec{r}'\\| &amp;= \\|\\vec{a} \\times \\vec{v}\\| \\\\&amp;= a v \\sin\\theta \\\\&amp;= \\frac{v^3}{\\rho},\\end{aligned}" />

where we used the cross product length formula <a
href="rvv.html#rvv-el">#rvv-el</a> and equation <a
href="/dyn/vector_calculus#rvv-el">#rvv-el</a> and equation <a
href="#rkt-er">#rkt-er</a> for the radius of curvature
<InlineEquation equation="\\rho" />. By definition <a href="#rkt-ek">#rkt-ek</a> the
curvature is <InlineEquation equation="\\kappa = 1/\\rho" />, so
Expand Down Expand Up @@ -1218,12 +1218,12 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<DisplayEquation title="Curvature of an explicitly defined function \\(y = f(x).\\)" id="rkt-e3" background="True" derivation="True" equation="\\kappa = \\frac{|y''(x)|}{(1 + y'(x)^2)^{3/2}}">
<p>
Use equation <a
href="rkt.html#rkt-e2">#rkt‑e2</a> and parametrize the curve using \(x\) as the parametrization variable. Namely:
href="#rkt-e2">#rkt‑e2</a> and parametrize the curve using \(x\) as the parametrization variable. Namely:

<DisplayEquation equation="\\begin{aligned}x = u, \\quad \\quad y = y(u) = y(x)\\end{aligned}" />

This yields a very elegant expression, as <InlineEquation equation="x'(u) = 1" /> and <InlineEquation equation="x''(u) = 0" />, which lets us arrive at the desired expression <a
href="rkt.html#rkt-e3">#rkt‑e3</a>.
href="#rkt-e3">#rkt‑e3</a>.
</p>
</DisplayEquation>
</SubSubSection>
Expand Down Expand Up @@ -1595,8 +1595,8 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<CalloutContainer slot="cards">
<CalloutCard title="Reference material">
<ul>
<li><a href="rkv.html">Position, velocity, and acceleration</a></li>
<li><a href="rkt.html">Tangential/normal coordinates</a></li>
<li><a href="#velocity_acceleration_vectors">Position, velocity, and acceleration</a></li>
<li><a href="#tangential_normal_basis">Tangential/normal coordinates</a></li>
</ul>
</CalloutCard>
</CalloutContainer>
Expand Down
8 changes: 4 additions & 4 deletions src/pages/dyn/particle_kinetics.astro
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Expand Up @@ -185,7 +185,7 @@ import DisplayTable from "../../components/DisplayTable.astro"
<p>
<strong>2. Kinematics:</strong> Using the polar
basis acceleration equation <a
href="rkv.html#rkv-ep">#rkv-ep</a> gives:
href="/dyn/particle_kinematics#rkv-ep">#rkv-ep</a> gives:
<DisplayEquation equation="\\begin{aligned} \\vec{a} &amp;= (\\ddot{r} - r\\dot\\theta^2) \\,\\hat{e}_r + (r\\ddot\\theta + 2\\dot{r}\\dot\\theta) \\,\\hat{e}_\\theta \\\\ &amp;= -\\ell \\dot\\theta^2 \\,\\hat{e}_r + \\ell\\ddot\\theta \\,\\hat{e}_\\theta. \\end{aligned}" />
</p>
<p>
Expand Down Expand Up @@ -446,8 +446,8 @@ import DisplayTable from "../../components/DisplayTable.astro"
<CalloutContainer slot="cards">
<CalloutCard title="Required material">
<ul>
<li><a href="rep.html">Kinetics of point masses</a></li>
<li><a href="reg.html">Kinetics of rigid bodies</a></li>
<li>Kinetics of point masses</li>
<li><a href="/dyn/rigid_body_kinetics">Kinetics of rigid bodies</a></li>
</ul>
</CalloutCard>
<CalloutCard title="Extras!">
Expand Down Expand Up @@ -611,7 +611,7 @@ import DisplayTable from "../../components/DisplayTable.astro"
class="seq-toggle:avb-fp-c:fbd" onclick="avb_fp_c.stepSequence('fbd')">free body diagram</button>
is visible. Now increase the speed of the bus to produce a
centripetal acceleration. <a
href="rkn.html#rkn-en">Newton's law</a> implies that there
href="/sta/introduction#rkn-en">Newton's law</a> implies that there
must be a centripetal friction force producing this
acceleration. As the speed increases, we will eventually
reach a point when the friction force required is too large
Expand Down
18 changes: 9 additions & 9 deletions src/pages/dyn/vector_calculus.astro
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Expand Up @@ -63,7 +63,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
product</em> or <em>scalar product</em>) is defined by
</p>

<DisplayEquation equation="\\vec{a} \\cdot \\vec{b}= a_1 b_1 + a_2 b_2 + a_3 b_3" title="Dot product from components." background="True"/>
<DisplayEquation equation="\\vec{a} \\cdot \\vec{b}= a_1 b_1 + a_2 b_2 + a_3 b_3" title="Dot product from components." background="True" id="rvv-es"/>

<p>
An alternative expression for the dot product can be
Expand Down Expand Up @@ -177,7 +177,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<DisplayEquation id="rvi-ed" title="Dot product symmetry." equation="\\vec{a} \\cdot \\vec{b} = \\vec{b} \\cdot \\vec{a}" background="true" derivation="true">
<p>
Using the coordinate expression <a
href="rvv.html#rvv-es">#rvv-es</a> gives:
href="#rvv-es">#rvv-es</a> gives:

<DisplayEquation equation=" \\vec{a} \\cdot \\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = b_1 a_1 + b_2 a_2 + b_3 a_3 = \\vec{b} \\cdot \\vec{a}." />
</p>
Expand All @@ -186,7 +186,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<DisplayEquation id="rvi-eg" title="Dot product vector length." equation="\\vec{a} \\cdot \\vec{a} = \\|a\\|^2" background="true" derivation="true">
<p>
Using the coordinate expression <a
href="rvv.html#rvv-es">#rvv-es</a> gives:
href="#rvv-es">#rvv-es</a> gives:

<DisplayEquation equation=" \\vec{a} \\cdot \\vec{a} = a_1 a_1 + a_2 a_2 + a_3 a_3 = \\|a\\|^2." />
</p>
Expand All @@ -195,7 +195,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<DisplayEquation id="rvi-ei" title="Dot product bi-linearity." equation="\\begin{aligned} \\vec{a} \\cdot(\\vec{b} + \\vec{c}) &amp;=\\vec{a} \\cdot \\vec{b} + \\vec{a}\\cdot \\vec{c} \\\\ (\\vec{a} +\\vec{b}) \\cdot \\vec{c} &amp;=\\vec{a} \\cdot \\vec{c} + \\vec{b}\\cdot \\vec{c} \\\\ \\vec{a} \\cdot (\\beta\\vec{b}) &amp;= \\beta (\\vec{a} \\cdot\\vec{b}) = (\\beta \\vec{a}) \\cdot\\vec{b}\\end{aligned}" background="true" derivation="true">
<p>
Using the coordinate expression <a
href="rvv.html#rvv-es">#rvv-es</a> gives:
href="#rvv-es">#rvv-es</a> gives:

<DisplayEquation equation="\\begin{aligned} \\vec{a} \\cdot (\\vec{b} + \\vec{c}) &amp;= a_1 (b_1 + c_1) + a_2 (b_2 + c_2) + a_3 (b_3 + c_3) \\\\ &amp;= (a_1 b_1 + a_2 b_2 + a_3 b_3) + (a_1 c_1 + a_2 c_2 + a_3 c_3) \\\\ &amp;= \\vec{a} \\cdot \\vec{b} + \\vec{a} \\cdot \\vec{c} \\\\ (\\vec{a} + \\vec{b}) \\cdot \\vec{c} &amp;= (a_1 + b_1) c_1 + (a_2 + b_2) c_2 + (a_3 + b_3) c_3 \\\\ &amp;= (a_1 c_1 + a_2 c_2 + a_3 c_3) + (b_1 c_1 + a_2 c_2 + a_3 c_3) \\\\ &amp;= \\vec{a} \\cdot \\vec{c} + \\vec{b} \\cdot \\vec{c} \\\\ \\vec{a} \\cdot (\\beta \\vec{b}) &amp;= a_1 (\\beta b_1) + a_2 (\\beta b_2) + a_3 (\\beta b_3) \\\\ &amp;= \\beta (a_1 b_1 + a_2 b_2 + a_3 b_3) \\\\ &amp;= \\beta (\\vec{a} \\cdot \\vec{b}) \\\\ &amp;= (\\beta a_1) b_1 + (\\beta a_2) b_2 + (\\beta a_3) b_3 \\\\ &amp;= (\\beta \\vec{a}) \\cdot \\vec{b}. \\end{aligned}" />
</p>
Expand Down Expand Up @@ -273,7 +273,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"

<DisplayEquation id="rvv-el2" title="Cross product length." equation="\\| \\vec{a} \\times \\vec{b} \\| = a b \\sin\\theta" background="true" derivation="true">
<p>
Using <a href="rvi.html#rvi-el">Lagrange's identity</a> we can calculate:
Using <a href="https://en.wikipedia.org/wiki/Lagrange%27s_identity" target="_blank">Lagrange's identity</a> we can calculate:

<DisplayEquation equation="\\begin{aligned} \\| \\vec{a} \\times \\vec{b} \\|^2 &amp;= \\|\\vec{a}\\|^2 \\|\\vec{b}\\|^2 - (\\vec{a} \\cdot \\vec{b})^2 \\\\ &amp;= a^2 b^2 - (a b \\cos\\theta)^2 \\\\ &amp;= a^2 b^2 (1 - \\cos^2\\theta) \\\\ &amp;= a^2 b^2 \\sin^2\\theta. \\end{aligned}" />

Expand Down Expand Up @@ -317,7 +317,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<DisplayEquation id="rvi-ea" title="Cross product anti-symmetry." equation="\\begin{aligned} \\vec{a} \\times\\vec{b} = - \\vec{b} \\times\\vec{a}\\end{aligned}" background="true" derivation="true">
<p>
Writing the component expression <a
href="rvv.html#rvv-ex">#rvv-ex</a> gives:
href="#rvv-ex">#rvv-ex</a> gives:

<DisplayEquation equation="\\begin{aligned} \\vec{a} \\times \\vec{b} &amp;= (a_2 b_3 - a_3 b_2) \\,\\hat{\\imath} + (a_3 b_1 - a_1 b_3) \\,\\hat{\\jmath} + (a_1 b_2 - a_2 b_1) \\,\\hat{k} \\\\ &amp;= -(a_3 b_2 - a_2 b_3) \\,\\hat{\\imath} - (a_1 b_3 - a_3 b_1) \\,\\hat{\\jmath} - (a_2 b_1 - a_1 b_2) \\,\\hat{k} \\\\ &amp;= -\\vec{b} \\times \\vec{a}. \\end{aligned}" />
</p>
Expand All @@ -335,7 +335,7 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
<DisplayEquation id="rvi-eb2" title="Cross product bi-linearity." equation="\\begin{aligned}\\vec{a} \\times (\\vec{b} + \\vec{c})&amp;= \\vec{a} \\times \\vec{b} + \\vec{a} \\times \\vec{c} \\\\(\\vec{a} + \\vec{b}) \\times \\vec{c}&amp;= \\vec{a} \\times \\vec{c} + \\vec{b} \\times \\vec{c} \\\\\\vec{a} \\times (\\beta \\vec{b})&amp;= \\beta (\\vec{a} \\times \\vec{b})= (\\beta \\vec{a}) \\times \\vec{b}\\end{aligned}" background="true" derivation="true">
<p>
Writing the component expression <a
href="rvv.html#rvv-ex">#rvv-ex</a> for the first
href="#rvv-ex">#rvv-ex</a> for the first
equation gives:

<DisplayEquation equation="\\begin{aligned} \\vec{a} \\times (\\vec{b} + \\vec{c}) &amp;= (a_2 (b_3 + c_3) - a_3 (b_2 + c_2)) \\,\\hat{\\imath} \\\\ &amp;\\quad + (a_3 (b_1 + c_1) - a_1 (b_3 + c_3)) \\,\\hat{\\jmath} \\\\ &amp;\\quad + (a_1 (b_2 + c_2) - a_2 (b_1 + c_1)) \\,\\hat{k} \\\\ &amp;= \\Big((a_2 b_3 - a_3 b_2) \\,\\hat{\\imath} + (a_3 b_1 - a_1 b_3) \\,\\hat{\\jmath} + (a_1 b_2 - a_2 b_1) \\,\\hat{k} \\Big) \\\\ &amp;\\quad + \\Big((a_2 c_3 - a_3 c_2) \\,\\hat{\\imath} + (a_3 c_1 - a_1 c_3) \\,\\hat{\\jmath} + (a_1 c_2 - a_2 c_1) \\,\\hat{k} \\Big) \\\\ &amp;= \\vec{a} \\times \\vec{b} + \\vec{a} \\times \\vec{c}. \\\\ \\end{aligned}" />
Expand Down Expand Up @@ -581,8 +581,8 @@ import InlineCanvas from "../../components/InlineCanvas.astro"
When we differentiate a vector by differentiating each
component and leaving the basis vectors unchanged, we
are assuming that the basis vectors themselves are not
changing with time. If they are, then we need to <a
href="rkv.html#rkv-sb">take this into account</a> as
changing with time. If they are, then we need to
take this into account as
well.
</p>
</Warning>
Expand Down
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