Mathematics - Mathematical Analysis Directional Derivatives by drifter1

mathematics · @drifter1 · Apr 18 '18

$25.47

Mathematics - Mathematical Analysis Directional Derivatives

<html>
<p><img src="https://i.stack.imgur.com/VOPuE.png" width="466" height="475"/></p>
<p><a href="https://mathematica.stackexchange.com/questions/27352/adding-tangent-velocity-vectors-to-a-plot-of-a-space-curve">Source</a></p>
<p><br></p>
<h2>Introduction</h2>
<p>&nbsp;Hello it's a me again drifter1!</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Today, we will continue with our <strong>Mathematical Analysis</strong> series of Mathematics by getting into <strong>Directional Derivatives</strong> that are based on <a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-partial-derivatives">Partial Derivatives</a> that we covered last time.</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As I already said last time, Directional Derivatives are used for <strong>finding normal vectors to a plane </strong>and also for <strong>finding tangent planes</strong>.</p>
<p>There is not much more to say about, so let's get started!</p>
<h2>Small recap</h2>
<p>Last time we talked about Derivatives of vector functions (curves) on top of planes.</p>
<p>If z = f(x, y) and r(t) = x(t)i + y(t)j then the <strong>derivative </strong><em><strong>along the</strong></em><strong> curve</strong> is given by:</p>
<p><em>f '(t) = df/dt = &nbsp;∂f/∂x * dx/dt + ∂f/∂y * dy/dt</em></p>
<p>We can see that we use the <strong>rule of composition</strong>, because those functions are "complex".</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;The function f(x, y) is a function of x and y, but x(t) and y(t) are functions of t. This means that we first partially differentiate f(x, y) by x or y and multiply this with the derivative of x(t) or y(t) with t.</p>
<p>Of course the derivative along a curve can also be found for 3 or even more dimensions...</p>
<p><br></p>
<h2>Tangent vectors</h2>
<p>Suppose the plane f(x, y, z) = c and the curve r(t) = x(t)i + y(t)j + z(t)k.</p>
<p>The derivative along the curve is:</p>
<p><em>u = df/dt = ∂f/∂x * dx/dt + ∂f/∂y * dy/dt + ∂f/∂z* dz/dt&nbsp;</em></p>
<p><br></p>
<p>The <strong>tangent vector of the curve r</strong> is:</p>
<p><img src="http://quicklatex.com/cache3/99/ql_060b20d8dd26b1568a29e901c86a3e99_l3.png"/></p>
<p><a href="http://quicklatex.com/cache3/99/ql_060b20d8dd26b1568a29e901c86a3e99_l3.png">quicklatex</a></p>
<p><br></p>
<p>The <strong>gradient of f</strong> (or Del of f) is the vector:</p>
<p>&nbsp;∇<em>f = gradf = (∂f/∂x, ∂f/∂y, ∂f/∂z)</em></p>
<p><br></p>
<p>So, the <strong>derivative along the curve</strong> is:</p>
<p><em>df/dt = (∂f/∂x, ∂f/∂y, ∂f/∂z) * (dx/dt, dy/dt, dz/dt) = </em>∇<em>f * u</em></p>
<p>where ∇<em>f is the gradient of f and u is a tangent vector of curve r(t) = (x(t), y(t), z(t))</em></p>
<p><br></p>
<p>If u =(a, b, c) is a unit vector (|u| = 1) and P0 a Point in space then:</p>
<p>∇<em>f(P0) * u = Duf(P0)</em></p>
<p>is called the <strong>derivative of f </strong><em><strong>along </strong></em><strong>the direction of u = (a, b, c)</strong>.</p>
<p><br></p>
<p>For <strong>example</strong>:</p>
<p><img src="http://quicklatex.com/cache3/2a/ql_6f4ca90c71aa8c0efd9f45a763ad0b2a_l3.png"/></p>
<p><img src="http://quicklatex.com/cache3/da/ql_5f3f908f3bf0a30edc51683b467e13da_l3.png"/></p>
<p><img src="http://quicklatex.com/cache3/f8/ql_6a63aae5ef346d3257a09da9d05a98f8_l3.png"/></p>
<p>Duf(P) = ?</p>
<p>The given vector u is not an unit vector and so we have to <em>convert </em>it.</p>
<p><img src="http://quicklatex.com/cache3/02/ql_2214cd126f9df59305271d33c303d902_l3.png"/></p>
<p><img src="http://quicklatex.com/cache3/43/ql_9414dd19ade221fb6dcd63d2f68dec43_l3.png"/></p>
<p>The gradient of f is:</p>
<p>∇f = (∂f/∂x, ∂f/∂y) = (2x + 2y, 2x - 6y)</p>
<p>For the point P we have:</p>
<p>∇f(P) = (1 + 1, 1 - 3) = (2, -2)</p>
<p>That way:</p>
<p>Duf(P0) = ∇f(P) * u0 =&gt;</p>
<p><img src="http://quicklatex.com/cache3/09/ql_5d33e630ef6e4296eb519db51a356509_l3.png"/></p>
<p>Some equations through <a href="http://quicklatex.com/">quicklatex.</a></p>
<p><br></p>
<h2>An explanation</h2>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Suppose we have the same plane and curve as before, the unit vector u = (u1, u2, u3) and the Point P0(x0, y0, z0).</p>
<p>And ε is a line that passes through P0, is parallel to u and has the parametric equations:</p>
<p><em>x = x0 + u1t</em></p>
<p><em>y = y0 + u2t</em></p>
<p><em>z = z0 + u3t</em></p>
<p>Then dx/dt = u1, dy/dt = u2, dz/dt = u3 and so:</p>
<p>df/dt = ∇f(P0) * (u1, u2, u3) = ∇f(P0) * u = Duf(P0)&nbsp;</p>
<p>Which of course is the derivative of f along the direction of u.</p>
<p><br></p>
<h2>Some interesting cases</h2>
<p>We can write the same equation like that:</p>
<p><em>Duf(P0) = ∇f(P0) * u = |∇f(P0) ||u|*cosφ</em></p>
<p>where φ is the angle between the vectors ∇f(P0) and u.</p>
<blockquote>This is clearly a "dot" product between vectors...</blockquote>
<p><br></p>
<p>Depending on the angle we have the following <strong>cases</strong>:</p>
<ol>
  <li>φ = 0, ∇f(P0) and u are parallel =&gt; Duf(P0) = |∇f(P0)| (highest possible increase of f)</li>
  <li>φ = π, ∇f(P0) = -u =&gt; Duf(P0) = -|∇f(P0)| (highest possible decrease of f)</li>
  <li>φ = π/2, ∇f(P0) and u are across and so Duf(P0) = 0</li>
</ol>
<p>And so:</p>
<p><em>-|∇f(P0)| &lt;= Duf(P0) &lt;= |∇f(P0)|</em></p>
<p><br></p>
<h2>Normal vectors to a plane</h2>
<p>So, now after all that we can finally get into the actual useful thing of this post...</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Suppose S: f(x, y , z) = c being a plane/surface and r(t) = x(t)i + y(t)j +z(t)k a curve on top of it that passes through the point P0(x0, y0, z0).</p>
<p>u = dr/dt = (dx/dt, dy/dt, dz/dt) is a tangent vector of curve r at P0.</p>
<p><br></p>
<p>Let's also suppose N = (a, b, c) being a <strong>normal vector to our plane</strong>.</p>
<p>Of course we can find a ∇f(P0) so that:</p>
<p>∇f(P0) * u = 0, which means that they are vertically across..</p>
<p>So, <strong>N = ∇f(P0)&nbsp;</strong></p>
<p><br></p>
<h2>Tangent planes</h2>
<p>By using this concept, planes/surfaces of the form:</p>
<p><em>a(x - x0) + b(y- y0) + c(z - z0) = 0</em></p>
<p>have N = (a, b, c) as a normal vector and also pass through P0.</p>
<p><br></p>
<p>So, with f(x, y, z) = c being a plane and P0(x0, y0, z0) a point,</p>
<p>the <strong>equation of a tangent plane</strong> is:</p>
<p><em>∂f/∂x (x - x0) + ∂f/∂y (y - y0) + ∂f/∂z (z - z0) = 0</em></p>
<p><br></p>
<p>For a surface f(x, y) = z we have:</p>
<p>F(x, y, z) = f(x, y) - z = 0</p>
<p>And so;</p>
<p><em>∂F/∂x = ∂f/∂x</em></p>
<p><em>∂F/∂y = ∂f/∂y</em></p>
<p><em>∂F/∂z = -1</em></p>
<p>Which means that:</p>
<p>N = ∇f(P0) = (<em>∂f/∂x, ∂f/∂y, -1</em>)</p>
<p>And finally the equation of such a tangent plane is:</p>
<p><em>∂f/∂x (x - x0) + ∂f/∂y (y - y0) - (z - z0) = 0</em></p>
<p><br></p>
<p><br></p>
<p>And this is actually it for today and I hope that you enjoyed it!</p>
<p>Next time we will start getting into double/surface integrals!</p>
<p>Bye!</p>
</html>

👍 zahnspange, hendrikdegrote, knozaki2015, trumpman, b0y2k, romedog, anyx, justtryme90, steemstem, lafona-miner, aizensou, yoshiko, nextgencrypto, pharesim, abigail-dantes, ozchartart, richman, thecyclist, meerkat, joseph, platinum-blue, thebluepanda, curie, eveuncovered, steemychicken1, themarkymark, sirwinchester, ssekulji, hagie, ruth-girl, sherlockcupid, razvanelulmarin, anwenbaumeister, minimalpris, bigdaddy, dannystravels, kushed, kinakomochi, ngc, lemouth, stitchybitch, i-gordan, derrick829, toyman, sunshinetraveler, drifter1, kerriknox, rxhector, yougotflagged, steem.services, timbernana, thedelegator, arbitrarykitten, steemservices, stoner19, spg, thisisbenbrick, dream-o, anarchyhasnogods, biancajapan, einsteinpotsdam, jerryblanceton, liberosist, koskl, and 118 others

`author`	drifter1
`permlink`	mathematics-mathematical-analysis-directional-derivatives
`category`	mathematics
`json_metadata`	{"tags":["mathematics","analysis","derivatives","steemstem","steemiteducation"],"image":["https://i.stack.imgur.com/VOPuE.png","http://quicklatex.com/cache3/99/ql_060b20d8dd26b1568a29e901c86a3e99_l3.png","http://quicklatex.com/cache3/2a/ql_6f4ca90c71aa8c0efd9f45a763ad0b2a_l3.png","http://quicklatex.com/cache3/da/ql_5f3f908f3bf0a30edc51683b467e13da_l3.png","http://quicklatex.com/cache3/f8/ql_6a63aae5ef346d3257a09da9d05a98f8_l3.png","http://quicklatex.com/cache3/02/ql_2214cd126f9df59305271d33c303d902_l3.png","http://quicklatex.com/cache3/43/ql_9414dd19ade221fb6dcd63d2f68dec43_l3.png","http://quicklatex.com/cache3/09/ql_5d33e630ef6e4296eb519db51a356509_l3.png"],"links":["https://mathematica.stackexchange.com/questions/27352/adding-tangent-velocity-vectors-to-a-plot-of-a-space-curve","https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-partial-derivatives","http://quicklatex.com/cache3/99/ql_060b20d8dd26b1568a29e901c86a3e99_l3.png","http://quicklatex.com/"],"app":"steemit/0.1","format":"html"}
`created`	2018-04-18 20:04:33
`last_update`	2018-04-18 20:04:33
`depth`	0
`children`	1
`last_payout`	2018-04-25 20:04:33
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	20.094 HBD
`curator_payout_value`	5.374 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	6,874
`author_reputation`	98,202,866,830,354
`root_title`	"Mathematics - Mathematical Analysis Directional Derivatives"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	50,818,076
`net_rshares`	4,781,585,246,582
`author_curate_reward`	""

properties (23)vote details (182)

voter	rshares	pct
berniesanders	4,028,067,512	2.5%
pharesim	76,915,018,414	0.15%
lafona-miner	105,243,900,737	5%
kushed	18,051,997,626	0.89%
nextgencrypto	78,796,302,486	2.5%
steemychicken1	42,762,458,128	25%
steemservices	10,153,464,676	2.5%
joseph	54,692,561,883	25%
aizensou	94,895,881,182	25%
b0y2k	278,869,756,769	25%
stoner19	9,803,007,717	25%
drifter1	12,149,003,000	100%
richman	68,629,350,553	25%
justtryme90	143,188,185,756	5%
eric-boucher	1,962,205,270	0.44%
anwenbaumeister	25,866,089,948	0.89%
razvanelulmarin	27,440,523,325	25%
rxhector	11,947,697,626	100%
anyx	195,306,425,231	20%
raymondspeaks	3,178,442,807	25%
knozaki2015	471,973,125,799	25%
shawnamawna	6,942,973,577	25%
liberosist	7,061,330,518	0.89%
hagie	30,866,685,363	25%
coinbar	4,691,282,505	25%
ozchartart	73,345,237,588	2.5%
thebluepanda	48,171,197,649	25%
yoshiko	84,066,087,199	25%
thisisbenbrick	8,852,114,121	25%
sirwinchester	37,876,658,376	25%
kiddarko	4,981,478,070	5%
lemouth	14,817,905,059	3.5%
rjbauer85	278,172,761	5%
anarchyhasnogods	8,465,368,312	2.5%
einsteinpotsdam	7,753,060,603	25%
dadview	1,247,865,085	3%
zahnspange	592,737,220,985	25%
thecyclist	64,423,263,245	2.5%
steemsquad	2,581,746,761	25%
allesgruen	2,781,519,465	25%
dannystravels	18,061,916,098	25%
jerryblanceton	7,271,864,137	25%
platinum-blue	50,478,892,422	25%
koskl	6,976,743,144	100%
meerkat	60,037,474,489	0.89%
ssekulji	34,266,937,198	25%
timbernana	10,609,865,003	25%
curie	47,443,919,495	0.89%
sherlockcupid	27,516,808,068	25%
hendrikdegrote	516,027,101,501	0.89%
dark.horse	5,139,933,262	25%
steemstem	125,769,324,954	5%
dashfit	114,456,221	0.44%
ejhaasteem	97,717,700	10%
adventureevryday	4,436,897,015	10%
sethroot	62,022,204	0.08%
toyman	13,487,047,005	25%
foundation	486,890,417	5%
the-devil	650,148,946	5%
i-gordan	14,381,446,924	25%
dna-replication	1,402,184,464	5%
boynashruddin	328,994,962	5%
yougotflagged	11,255,549,309	2.5%
buzzbeergeek	4,511,763,270	10%
pacokam8	84,331,429	0.22%
borislavzlatanov	429,320,521	5%
bigdaddy	22,089,440,057	25%
kingsmind	4,740,886,085	25%
jamhuery	499,070,664	5%
steemitcitizen	411,888,646	5%
dbzfan4awhile	882,222,566	0.75%
romedog	266,532,292,692	25%
sunshinetraveler	12,773,542,078	25%
eveuncovered	43,023,944,999	25%
kryzsec	2,286,624,994	5%
markangeltrueman	335,478,603	0.13%
spg	8,916,627,867	25%
freefuture	6,623,230,273	25%
kinakomochi	17,747,996,872	25%
tantawi	74,395,438	0.89%
trumpman	356,868,690,945	100%
samminator	1,981,952,687	5%
derrick829	14,282,488,581	1%
locikll	422,996,446	1.78%
dber	1,739,329,155	5%
aboutyourbiz	210,930,121	0.89%
kerriknox	12,126,653,432	5%
howtostartablog	189,123,325	0.04%
sammosk	5,553,869,365	25%
arbitrarykitten	10,264,894,121	25%
biancajapan	8,247,007,456	25%
slickhustler007	50,357,649	0.44%
ruth-girl	29,694,501,485	20%
ertwro	2,067,201,391	5%
thedelegator	10,568,327,881	2.5%
juanjdiaz89	211,774,647	5%
thinknzombie	1,566,525,272	0.44%
nitesh9	1,250,507,772	5%
stitchybitch	14,778,255,511	25%
churchboy	818,119,344	5%
himal	346,906,445	5%
themarkymark	41,615,834,957	25%
abigail-dantes	75,621,976,129	5%
leczy	294,643,650	5%
steemulator	293,466,522	0.89%
suravsingh	59,409,543	5%
misterakpan	102,370,496	0.04%
mountain.phil28	2,189,200,751	25%
akeelsingh	196,135,717	5%
mountainwashere	1,858,489,866	5%
avesa	391,901,390	0.1%
felixrodriguez	123,101,898	2.5%
minimalpris	22,168,652,549	100%
pearlumie	1,907,536,483	5%
gabox	177,542,242	0.04%
ostdait	66,913,801	100%
tsyur	397,768,917	100%
djlethalskillz	709,154,611	5%
teneiced	4,592,572,654	25%
infinitelearning	155,532,055	2%
massivevibration	3,532,072,037	5%
onartbali	431,606,604	5%
sumayyahsaidso	4,215,751,734	25%
clweeks	60,539,624	0.35%
torico	294,057,887	0.13%
ksolymosi	979,353,417	5%
simplifylife	151,006,178	0.1%
kriptonoob	718,482,603	1%
jordanx2	59,442,625	0.44%
playitforward	1,522,879,052	1.5%
thatterrioguy	55,222,578	0.66%
imamalkimas	61,206,955	0.89%
zeeshan003	107,270,476	5%
erodedthoughts	694,722,626	0.75%
peaceandwar	162,053,617	0.44%
enzor	85,444,733	2.5%
bobdos	299,457,130	0.08%
ngc	15,292,415,603	2.5%
carloserp-2000	833,062,737	5%
pangoli	435,566,337	5%
rachelsmantra	214,559,082	5%
gra	1,329,725,422	5%
skycae	241,161,816	0.89%
williamsq	513,600,623	100%
kenadis	1,219,881,233	5%
amavi	780,751,719	1%
robotics101	200,158,045	4%
gentleshaid	1,157,760,158	5%
cgbartow	173,381,951	2.5%
sco	185,500,025	0.25%
adetola	184,807,431	5%
rharphelle	1,958,026,388	100%
dysfunctional	226,824,101	2.5%
wanderingdanish	258,821,621	1%
whileponderin	122,351,384	5%
bennettitalia	96,293,790	0.04%
hadji	243,120,157	5%
sakura1012	137,667,389	5%
terrylovejoy	679,907,755	2%
wisewoof	109,841,084	0.44%
rionpistorius	98,027,620	2.5%
deutsch-boost	363,873,128	20%
wdoutjah	52,759,054	0.44%
dexterdev	937,013,865	5%
ugonma	72,451,024	5%
benleemusic	1,350,950,859	0.04%
chimtivers96	61,757,984	0.89%
sissyjill	73,923,961	7%
positiveninja	150,901,492	0.44%
theunlimited	52,227,985	10%
chillingotter	102,300,994	0.44%
dream-o	8,520,918,592	25%
pseudojew	84,046,426	5%
victorcovrig	69,961,057	1%
de-stem	1,801,671,454	4%
linkyourlife	3,591,319,054	25%
milagros07	233,062,064	100%
chloroform	313,979,708	5%
steem.services	10,782,154,524	1%
qanon1111	312,243,683	100%
niouton	129,006,750	0.17%
biomimi	180,496,636	40%

`author`	greentomorrow
`permlink`	re-drifter1-mathematics-mathematical-analysis-directional-derivatives-20180418t211422995z
`category`	mathematics
`json_metadata`	{}
`created`	2018-04-18 21:14:21
`last_update`	2018-04-18 21:14:21
`depth`	1
`children`	0
`last_payout`	2018-04-25 21:14:21
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	0.000 HBD
`curator_payout_value`	0.000 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	107
`author_reputation`	-132,273,054,154
`root_title`	"Mathematics - Mathematical Analysis Directional Derivatives"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	50,826,806
`net_rshares`	-6,979,429,071
`author_curate_reward`	""