line integral

corbalgasterthehybrid  asked:

In regards to that last question what if instead of one child falling down while Chara was a child all of the fallen children fell down during that timespan

(how dare you know that’s what I want to draw)

((Please note: Completely impossible AU, and I’m writing it so… hahahaha ))


Dear Mr. Benedict Cumberbatch,

Your fans stand with you.

Whether the Current War comes out this Thanksgiving or in 2018, I can say with complete certainty that we are incredibly proud to be your fans. You spoke up against Weinstein when people with less on the line have remained silent.

Your integrity, your ability to be articulate when facing the incomprehensible, is astonishing.

So if by some miracle you see this, know that we are 100% behind you.

And if you are concerned about the negative affects of the movie’s release being delayed on the movie or on your career, let me put your mind at ease.

Your talent speaks for itself. But if you really are worried, I can launch a campaign (at your request) to start conversations regarding your statement and the movie, and the monstrous actions and allegations that are surfacing.

Mr. C, your sense of responsibility and willingness to use your platform to address real issues will carry you forward, always.

With love and respect,


‘HORiZON’ is a visual novel rewrite of the loo season 3 in its entirety.

Horizon follows the point of view of the protagonist, Clarke Griffin, in the immediate aftermath of season 2 of ‘the 100’

Horizon is a choose your own adventure, with separate story lines and arcs, that integrate side plots and other exciting features giving readers a new and individualized experience with each new episode.  

This free –not for profit– work will be available via APP and our website.  

How can I get involved?

If you’re interested in getting involved, in whatever capacity, send us an email/ask/message with your interests and talents.

This isn’t a huge project and we want people who are willing to volunteer their time to be aware that this current project is already in progress and thus we have specific needs and assignments for you.

Right now, we are looking for a few talented artists to help alleviate the burden on our artists, who are AMAZING, but INCREDIBLY overworked.

What if I don’t have an artistic talent or much time? 

REBLOG AND SHARE our posts. This is incredibly important.Liking a post contributes to notes, re-blogging spreads awareness!

Visit us at 

Study With Me: Line Integrals

Hey guys! I’m currently studying for the Mathematics Subject Test of the GRE, which I plan on taking in the fall. One of the ways I like to study is by explaining the material to someone else. I currently have weekends off from research, and since Saturdays are for the boys, it leaves Sundays for GRE preparation. 

Because of this, every Sunday, I’ll explore a different undergraduate topic that could appear on the Mathematics Subject Test. This week: Line Integrals.

I’ll talk about the following:

  1. What is a line integral?
  2. How do you calculate a line integral?
  3. An Example

As a brief note, this post contains LaTeX code and will be much easier to read when viewed directly on my blog, where the code will compile!

What is a line integral?

Let’s first recall what we already know about integrals. We’re used to integrating functions of one variable over an interval [a,b]. We can think of this as integrating over the path on the x-axis from a to b, and the value of the integral as giving the area bounded by the curve y=f(x) over the path [a,b]. 

But! We can also integrate over paths that aren’t just straight lines along the x-axis. The resulting integral is called a curve, contour, or path integral. Most commonly, it is known as a line integral. 

In this post, I’ll be talking about line integrals with respect to arc length. 

Before we get into it, I’d like to start by defining what it means for a curve to be smooth. A curve, C, with parameterization r(t) = <x(t), y(t)> is smooth if the derivative r’(t) is continuous and nonzero. Additionally, we can say C is piecewise smooth if it is composed of a finite number of smooth curves joined at consecutive endpoints. Basically, this means I have a bunch of curves $C_1, C_2, …, C_n $ that are all individually smooth and Each $C_i$ has its endpoints connected to $C_{i-1}$ and $C_{i+1}$. 

Back to line integrals. Suppose we have a function f(x,y) and a smooth curve, C, in the x-y plane. We want to think about breaking C into n tiny pieces of arc length $\Delta s_i$. For each of the tiny pieces of C, choose any point $P_i = (x_i, y_i)$ and then multiply $f(P_i) = f(x_i, y_i)$ by the length $\Delta s_i$. This process is fairly similar to how we define integration for the case where the path is a line on the x-axis. We want to sum up these multiplied terms for all n terms. If the value of that sum approaches a finite, limiting value as $n \rightarrow \infty$, then the result is the line integral of f along C with respect to arc length. Below is a comparison of the single variable case integrating over a path [a,b] on the x-axis (left) and the line integral with respect to arc length over the curve C (right). 

Note the notation used for the line integral. If we’re integrating over a path C, we write C at the bottom of the integral. 

What does this mean geometrically?

The value of this integral is the area of the region whose base is C and whose height above each (x,y) point is given by f(x,y)

How do we actually calculate the line integral?

First, parameterize C. That is, for a parameter t, find the equations x=x(t) and y=y(t)  for $a \leq t \leq b$. We consider C to be directed, which means we’re saying that we trace C in a definite direction, which is called the positive direction. Basically, we’re saying that t runs from a to b, so A = (x(a), y(a)) is the initial point and B = (x(b), y(b)) is the final point. 

Since we have $(ds)^2 = (dx)^2 + (dy)^2$ (think Pythagorean theorem), we can write:

$\frac{ds}{dt} = \pm \sqrt{( \frac{dx}{dt})^2+(\frac{dy}{dt})^2}$

which can then be rewritten as:

$\pm \sqrt{(x’(t))^2+(y’(t))^2}$.

We use the + sign if the parameter t increases in the positive direction on C and the - sign if t decreases in the positive direction on C

So, we have:

$\int_C f \,ds = \int_{a}^{b} f(x(t), y(t))\frac{ds}{dt} \,dt$.

An Example

Determine the value of the line integral of the function f(x,y) = x + y^2 over the quarter-circle x^2 + y^2 = 4 in the first quadrant, from (2,0) to (0,2).

Solution below.

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“He is by no means a small role, in fact, he’s one of the most complex and gripping characters throughout. In fact, I’m pretty sure Harry Styles has the most lines and he is integral to very big plot developments. Alex is a full, rounded character.”

Kim & Briana meet and greet

So I know what is said at meet and greets is supposed to be sort of kept there. I would never relay something that seemed private or personal. But Kim & Bri spent most of the time talking about their current projects with so much energy and enthusiasm, I wanted to share with you guys to keep the excitement about Wayward Sisters amping up.

About Wayward Daughters:

Kim reiterated AGAIN how none of this is because of her and/or Bri. It’s all the fandom. She talked about how she feels such a strong obligation to the fandom to deliver every single time, whether it’s on stage at cons or on screen. She knows what it’s like to pinch every penny and knows how much cons cost and wants to give us everything. Plus she talked about how much she gets back from us with the love we pour her way. 

Then she and Bri both talked about how things are going for the new show. Right now they are mostly in training mode - learning lots of cool fighting techniques so they can be badass! They got a new fight choreographer from the show Lucifer (I think). He has a different method which the girls both really like. Before he began anything, he met with them and talked with them about the characters and how they thought their styles might be. Kim said her thought about Jodi is that this lady has seen it all. She’s a cop, is a little older, and is just over bullshit so she’s not going to stab or punch if she can just shoot. There was also a scene where she was supposed to scratch a monster’s face and they talked about it and decided Jodi is a puncher, not a scratcher (and they also discovered that Kim is a horrible kicker LOL). He also does this thing where he has stunt men film the whole fight scene first then the director edits and the girrls can watch it before they ever have to film. It helps to get a great end product and keep everyone safe. 

I asked about the writing and if they got any familiar writers. They, of course mentioned Bobo. They also said there have been a bunch of people on the SPN staff who are so excited about what this new show is doing, they have expressed interest in moving over, which really says a lot if they are willing to leave a very secure job and take a risk on something that may or may not even get picked up. We also all (including Kim & Bri) fangirled over Robbie Thompson and sent up a collective prayer to the universe that he decides to write for Wayward Sisters.

We also talked about how the casting is coming along. We all already knew about one of the regulars being the granddaughter of Missouri Mosely. They are also developing a native American character who’s character and story line is integral to her heritage. They didn’t want diversity to be a check box. They wanted an authentic and new perspective that we haven’t seen before so bringing in the supernatural aspects of native American culture should be really cool! They kept talking over and over again about how they are working on an authentic story while really trying to deliver on what the fans want. In fact, Kim acknowledged that the whole Charlie debacle actually made TPTB take a step back and rethink. She talked about how much fans care about these characters and their stories and especially someone like Charlie who felt like a representation of us and to kill her in a cheap way just for man pain was a poor decision that really backfired. They are trying to not make the same mistakes for WD. 


They talked about how incredibly hard gishwhes was and that they couldn’t believe the creativity and work that people put into it. Kim talked about one of the items she did that was her favorite even thought it broke the rules. It was the toy item and she had a ball making her toys do all sorts of naughty things that were too risque to submit. She said she kept thinking she should be able to submit them because every one of the bad behaviors was based on something Misha has told her he has done. ROFL!!!!

Briana’s album:

I asked Briana about her kickstarter. She lit up and told us that she never ever thought that making an album would be in her future. But after having the courage to step out on stage at the SNS, she couldn’t believe how much support she got from the fans. Then Jason Manns told her he would help her record an album. She is working on selecting songs now. It will be mostly covers that highlight her voice and that she can perform over an over because people will want to hear them. He is also encouraging her to write at least one of her own songs. 

Their whole m&g was pure joy! I will 100% splurge for this again next year if they are at PittCon. I just love these ladies so much and let me tell you, the feeling is definitely mutual. 


I’m half-way finished making chapter 1!

Feels like slow going cause I’m still figuring stuff out & learnin - and can’t be happy with my work, yet. I haven’t felt this in a while, but with an old project like this it seems like it always feels better while you’re dreaming about it, and hard to replace those perfect vague ideas;; 

But anyway - feels good at least to be learning! Gonna get there slowly but surely.

i’m lowkey in love with this structure where over the next week, we’re going to enter each character’s pov and come to understand more about them by the end of that clip, so that when we see them all together in the last clip at sana’s party, we’re going to see them and their group dynamic in a whole new light. they’ll still be the same people, but we’ll see them differently in the best way. i think that’s why the writers had eva’s party, another point where everyone was together, so that we had this parallel to compare it to. eva’s party and sana’s party will serve as this “before and after” idea whereby we are shown what happens to your perspective and understanding once you learn and understand people’s battles and backgrounds.

octopisunsets  asked:

Okay but Lena wanting to make it a surprise on date night so she consults Clark and has him use the Fortress' computer for patterns and it ends up being this perfectly clingy super low back number and Kara turns around and sees her and crushes the glass she's holding her hand

Could you imagine if that’s how she proposes to Kara? She goes to the Fortress with Clark and not only gets the patterns for the dress but learns about the traditional Kryptonian proposals and wedding ceremonies. 

Leading up to it, she practices the few lines of Kryptonese to integrate it into her speech. Finally, she gets Alex to sneak her into the DEO to ‘speak’ with her parents and ask for their blessing. Lena is fully aware that the holograms will not be able to answer appropriately but she still wanted to stand before Kara’s parents. 

This is something that I need to see like now! Or just Lena in that super sexy dress would work too. Maybe a wonderful artist like @lordsantiagoaz would draw a sketch? 

#236 - For emmernems & anonymous x2

Filling the prompts “…the idea of Van and his gf/wife/etc etc not being able to have kids. Like they’re trying for a baby but it’s just not happening and BAM! TESTTUBE BABY 2.0! They try IVF and the miracle of baby McCann is born, a quite literal mini Van.” from @emmernems and “it’s modern day and reader casually sings lyrics to Van from his songs that relate to their situation.”  and “reader is super into health and fitness and is a total gym bunny?”

Note: Shout out to @simpaticoswift - the most beautiful test tube baby ever.

Each store bought pregnancy test was discarded in the public trash can on the corner of your street. There was no use in both you and Van having constantly shattered hearts. The stupid plastic sticks all said the same thing, like a tragic groundhog day. Maybe they were wrong. Maybe dozens of them were wrong. Maybe every brand you had purchased over the course of a whole year and a bit were dead wrong.

In the beginning, neither of you were worried. But fear set in in you first, then Van. He’d watch you from the other side of the room pretending you really did care about the positioning of throw pillows and that all the jars in the fridge faced forward. He’d bite his nails and huff to himself, before escaping the sad to have a smoke alone outside. 

Nobody wanted to be the one to bring it up, make it real. Van was a true optimist and you knew he’d just keep hoping and trying before he was willing to admit that his family dream wasn’t going to pan out like all his others. You were simply terrified.

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anonymous asked:

Hello! I am an avid Destiel shipper, and I love your meta (Destiel and otherwise)! I am glad that supernatural has reached the point where they can have queer characters that don't die (what happened to Charlie still pisses me off though). However, as obvious as Dean and Cas' love and possible future relationship is to me, it's not so obvious to others, and I sincerely doubt that they'll ever do anything about it. If something happens in Season 13, what do you think they'd do? Thanks! <3

Hello my lovely!

Thank you so much, I’m very happy to read you’re enjoying my meta! (makes my heart go an extra big thump-thump!)

And I do apologise for this reply taking forever. I get inspired and feel I want to write a long response and then I know it’ll take half a day so then I postpone and I shouldn’t and, again, my apologies. 

I love this question and have been pondering it since I got it - as well as working my way through some meta posts which I suppose goes someways to telling you what I think they might do. But let me focus my thoughts.

Firstly, I understand that doubt is there, because there have been so many times over the years when they’ve suddenly pulled back on the Destiel of it all.

Dean and Cas have been separated by:

  • Delusions of grandeur (Cas in cahoots with Crowley, to then believe he’s God)
  • Death (Cas returning the souls to Hell, releasing the Leviathan)
  • Mental illness (Cas healing Sam)
  • A need for redemption (Cas choosing Purgatory)
  • Gadreel (human!Cas and Dean could not get cosy in S9)
  • Self-sacrifice (Cas saying yes to Lucifer)
  • Self-defence (neither thinking the other loves them for all of S12, Dean acting defensive as fuck, shutting Cas out because he wants to let him in so bad it hurts)
  • MISCOMMUNICATION (a big theme throughout their courtship)

So, here’s what I think S12 has built towards - open communication.

This doesn’t just go for Dean and Cas, though. It goes for all three of Team Free Will, because I feel TFW has been slowly moving towards assembly for all of S12, and I want them together, working together, planning together in S13.

If Cas wakes up human - either in that sunrise that seems to be promising to peak over the mountains by that lake where he died (which would be glorious, but which I doubt, simply because it’s too soon for him to come back given the magnitude of the REBIRTH that seems unavoidable here), or when crawling out of his grave after the brothers bury him - human!Cas will facilitate the Destiel story line becoming integrated into the surface narrative of the show.

To me this feels the most logical, but please know that I might be so so wrong.

Why does it feel logical?

Because the surface viewers (or the casual viewers), as you say, do not see the romantic love between Dean and Cas.

So what to do?

They will need to build the romance up from scratch to make it believable to the casual viewer. 

How to do that then?

Well, how about we position all three of our main characters on a threshold, where all of their arcs have come to a peak and is ready to move into a new beginning for each of them.

  • Sam will Lead and MoL
  • Dean will tear down more walls while letting Sam Lead and MoL
  • Cas will be reborn, with all that this rebirth will entail


New Beginnings.

The writers can opt to move into S13 with all they have set up in S12 - all that chaos and calamity that will work as a pressure wave of deepened threat and uncertainty in S13 - and anchor us in the new leaf each of our main characters are turning. It will make the season feel fresh and interesting. 

Either way, though, whatever happens, I know they’ll make it so damn awesome I can’t wait to see what they have planned for us!

To my mind, human!Cas would feel like Cas, but there were always things Castiel couldn’t partake of fully, that human!Cas can: food, drink, breathing the air, wanting to be alive, learning what that want means, falling properly in love, feeling longing, and misery, and longing for Dean, and carrying that overwhelming fear over from Castiel’s heart, that his love isn’t reciprocated.

Castiel has felt frustration. He’s felt sadness. He’s felt hopelessness. But he hasn’t been able to deal with these emotions because “It’s all so confusing” as he says in S11 after finding Dean’s porn on Dean’s laptop. (exclusively of women, of course) All of it’s always been so confusing to Cas. He feels so many things, but he can’t quite make heads or tails of his feelings. 

So this is something the writers would tackle, humanising him, grounding our beloved angel so that he can finally make the choice of where he wants to belong, not where he thinks he should belong or could belong or has to belong.

As for Dean, he’ll just be so damn happy having Cas back that I can see him following him around to make sure he’s ok, being overly attentive, unable to hide his emotions, and the casual viewer will take note. As they took note of the mixtape in 12x19. 

These gestures, if they make Destiel part of the surface narrative, and I can’t see why they would backtrack (seriously, the love story arc is peaking along with the individual character arcs addressed above so for them to backtrack now would hurt the narrative they’ve been building for so many years) (they won’t do that), these gestures will then cement the fact that Dean and Cas are in love with each other

Then again, Cas might come back a powerful angel. *shrug* We have no way of knowing for sure, of course!

But if they do what I believe they’re about to do, the writers will weave Destiel into the surface narrative slowly, out of necessity, so that the surface viewers who really couldn’t care less will just shrug and go, oh, okay, so they’re doing this now, while giving the middle finger to the viewers who gag and walk away. (walk away gaggers) (good riddance)

This is partially why I feel pretty confident that Cas will be back early in S13. Because I do so firmly believe in the love story of it all, and I do understand the precarious situation the writers are in of introducing it into the surface narrative.

It will take time, and many episodes, and a fresh feel about it, as with every other arc poised for New Beginnings in S13.

The mixtape was the first thread for Destiel to be woven into the surface, though.

It was the first visual manifestation of there being anything more between these men than brotherhood and friendship. (ok so I would say there have been about one thousand of these) (um trench coat in the trunk of Dean’s car for a year and handing it over) (yeah casual viewers) (SHAME on you)

All the other threads are just waiting, dangling on the wind, and oh will they make the prettiest tapestry EVER!

Unpopular opinion

But i’m sick of people wanting everything in s4 to revolve around Evak. I love Even and I adore Isak but this is Sana’s season and there are people in her life who are just as important to her that deserve screen time too.

Don’t get me wrong, I understand that both of them have story lines that are integral to the season, but that doesn’t mean that every time something happens and she needs to go running to Evak. 

Let her be her own hero.