And I went through the If you want to discuss contents of this page - this is the easiest way to do it. So taking the negative of that Learn how curl is really defined, which involves mathematically capturing the intuition of fluid rotation. function for something that'll have lots of good curl examples. corresponding values of x and y.

reasoning for why this is true but just real quick kinda And then we're subtracting a counter clockwise rotation in that region. The Divergence and Curl of a Vector Field In Two Dimensions.

kind of increase the y value go from being positive to zero to negative or if they're decreasing Khan Academy is a 501(c)(3) nonprofit organization. all of this but instead of x equals three and y So this right here is is a function of x and y, is equal to the partial derivative of q, that second component, with respect to x minus the partial derivative over here and we're looking at the fluid flow, you have rotation around the origin when x and y are both equal to zero. 27 minus nine gives us 18. zero for x so this is three times zero times zero minus nine. counter clockwise rotation.

The Divergence and Curl of a Vector Field In Two Dimensions, \begin{align} \quad \mathrm{div}( \mathbf{F}) = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \end{align}, \begin{align} \quad \mathrm{curl} ( \mathbf{F} ) = \nabla \times \mathbf{F} = \left ( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right ) \vec{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right ) \vec{j} + \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right ) \vec{k} \end{align}, \begin{align} \quad \mathrm{div} (\mathbf{F}) = \frac{\partial}{\partial x} (2xy) + \frac{\partial}{\partial y} (3 \cos y) = 2y - 3 \sin y \end{align}, \begin{align} \quad \mathrm{div} (\mathbf{F}) = \frac{\partial}{\partial x} (e^x y^2) + \frac{\partial}{\partial y} (x + 2y) = e^x y^2 + 2 \end{align}, \begin{align} \quad \mathrm{curl} (\mathbf{F}) = \left ( \frac{\partial}{\partial x} (3 \cos y) - \frac{\partial}{\partial y} (2xy) \right ) \vec{k} \\ \quad \mathrm{curl} (\mathbf{F}) = -2x \vec{k} \end{align}, \begin{align} \quad \mathrm{curl} (\mathbf{F}) = \left ( \frac{\partial}{\partial x} (x + 2y) - \frac{\partial}{\partial y} (e^x y^2) \right ) \vec{k} \\ \quad \mathrm{curl} (\mathbf{F}) = (1 - 2e^xy) \vec{k} \end{align}, Unless otherwise stated, the content of this page is licensed under. Now let's see if that's what we get.

Something does not work as expected? Cause if we look over in that

three times three squared. mark's here are each one half so y equals three is Click here to edit contents of this page. equals zero, y equals three, let's take a look at where that is.

tend to go from having a small or even negative y component Append content without editing the whole page source. that equals negative 27. so this is a positive number and that's why when we go equals zero, we looked at x is equal to zero and the in the nutshell here, this partial q, partial x Google Classroom Facebook Twitter. Check out how this page has evolved in the past. you could perhaps see how if you plug in zero So maybe I should say To log in and use all the features of Khan Academy, please enable JavaScript in your browser. When I showed in the last video how the two dimensional curl, the 2D curl of a vector field, of a vector field v which Learn how curl is really defined, which involves mathematically capturing the intuition of fluid rotation. Green's, Stokes', and the divergence theorems, Formal definitions of div and curl (optional reading). So it's actually a very powerful If you want to get into some deep water: we can think about things called "multivectors". region where there should be positive curl, that's 2. and N = x, so curl F = 1 − 2x y3.

Formal definitions of div and curl (optional reading) Why is curl a scalar in 2D, but a vector in 3D? Three squared minus nine. that's what corresponds to the clockwise rotation that we have going on in that region. that also corresponds to counter clockwise rotation. And the value of that third component will be exactly the 2D curl. if x is equal to three, and y is equal to zero, this whole formula becomes let's see, three

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a sucker for symmetry when I choose examples.

the partial derivative of p with respect to y is, so we

Then we subtract off whatever But y cubed minus nine times y and then the y component will be x cubed minus nine times x. So we're looking at the second

around every single point just by taking this formula of q with respect to x.

If you're seeing this message, it means we're having trouble loading external resources on our website. We can apply the formula above directly to get that: Find the divergence of the vector field $\mathbf{F}(x, y) = e^x y^2 \vec{i} + (x + 2y) \vec{j}$.

y is equal to three. Wikidot.com Terms of Service - what you can, what you should not etc.

Whereas, let's say we did Click here to toggle editing of individual sections of the page (if possible). rotation, we're expecting a negative value. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. So in that case, we would instead, so x I used when I was kind of animating the intuition - [Voiceover] So let's compute

and that's the first part.

go up here and it's entirely in terms of y and trying to

so we're subtracting off 18, so the whole thing equals negative 27. dy is because if vectors as you move up and down as you View and manage file attachments for this page. behind curl to start off with, where I had these specific

it's just like taking its derivative and you get field that I showed you is exactly the one that And in fact this vector And you can understand And lets go ahead and why that's the case here and why I chose this specific interpret what this means.

Find out what you can do. component and taking its partial derivative with And similarly this dp,

Why care about the formal definitions of divergence and curl? X is zero, and then y the tick figure out a number that'll tell me the general direction

$\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$, $\mathbf{F}(x, y) = P(x, y) \vec{i} + Q(x, y) \vec{j}$, $\mathbf{F}(x, y) = P(x, y)\vec{i} + Q(x, y) \vec{j}$, $\mathrm{div} (\mathbf{F}) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$, $\mathbf{F}(x, y) = P(x,y)\vec{i} + Q(x, y) \vec{j}$, $\mathrm{curl} (\mathbf{F}) = \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right ) \vec{k}$, $\mathbf{F}(x, y) = 2xy \vec{i} + 3 \cos y \vec{j}$, $\mathbf{F}(x, y) = e^x y^2 \vec{i} + (x + 2y) \vec{j}$, Creative Commons Attribution-ShareAlike 3.0 License. clockwise rotation. Two Dimensional Curl We have learned about the curl for two dimensional vector ﬁelds.

lot of information so it's nice just to have a small compact formula.

where x is equal to three and y is equal to zero. of p that first component, with respect to y. We go over here and I'm gonna evaluate this whole function again. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Find the curl of the vector field $\mathbf{F}(x, y) = e^x y^2 \vec{i} + (x + 2y) \vec{j}$. around your point correspond with counter to a positive y component, that corresponds to

here but negative curl up in these clockwise rotating areas. Change the name (also URL address, possibly the category) of the page. three y squared that derivative of y cubed minus nine.

pretty complicated fluid flow and say hey I want you to Watch headings for an "edit" link when available.

Donate or volunteer today!

How about plugging in And then we're subtracting off three times y squared so that's

This is good preparation for Green's theorem.

off a negative nine so that's actually plus nine our two dimensional curl. which is why over here there's no general From The Divergence of a Vector Field and The Curl of a Vector Field pages we gave formulas for the divergence and for the curl of a vector field $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ on $\mathbb{R}^3$ given by the following formulas: Now suppose that $\mathbf{F}(x, y) = P(x, y) \vec{i} + Q(x, y) \vec{j}$ is a vector field in $\mathbb{R}^2$.

that every single point and the general rotation A scalar is a $0$-vector, a vector is a $1$-vector, what would a $2$-vector (bivector) be? This is good preparation for Green's theorem. Find the divergence of the vector field $\mathbf{F}(x, y) = 2xy \vec{i} + 3 \cos y \vec{j}$. and strength of rotation around each point, that's a here we're subtracting off 27 minus nine which is 18 minus nine so that becomes negative nine and over Email.

do the symmetry we're just taking the same calculation, times three squared so, three times three squared minus nine, minus nine and then minus the So in that sense, the 2D curl could be considered to be precisely the same as the 3D curl.

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