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## Multivariable calculus

### Course: Multivariable calculus>Unit 5

Lesson 4: 2D divergence theorem

# Constructing a unit normal vector to a curve

Figuring out a unit normal vector at any point along a curve defined by a position vector function. Created by Sal Khan.

## Want to join the conversation?

• Calculating the normal unit vector seems pretty straightforward. What is the purpose of finding it? Why kind of applications would you use this in? • I realize it was two years ago but I'll answer the question anyways if someone is wondering. The purpose of a unit vector is to find the direction in which a vector is traveling in (its magnitude is one.) With this, you can manipulate it and other vectors to have them travel in same direction or different directions easier. A calculus IV concept I recently learned was with a gradient of a function, knowing that a unit vector in the same direction as the gradient would give the maximum possible change of the gradient.
• I am thinking of a proof of the unit normal vector being equaled to that expression using dot product. Actually, I have done it using mspaint lol.
Here is my proof:
http://postimg.org/image/nuj1gp3xb/
http://postimg.org/image/bjnislcmh/
I think it is fairly easy to do it and to understand it once you have learnt vector calculus. (Although It took me an hour of trial and error to figure out this proof, I tried using gradient to prove it but failed. ) May I ask if you want to or not to include this or any similar proofs in your video guides? I guess this will be helpful for the audience. • Isn't a tangent vector dr/dt rather than just dr? Why specifially dr/dt cannot be a tangent vector to a curve? • at ~, I thought that the normal vector always points in the direction the curve is curving. So wouldnt the unit normal vector point in the opposite direction (towards the origin)? • Why is -dxj going up when it should be going down since it is negative? • Does it really matter which component you make negative? There is a khan academy article on constructing unit normal vectors to curves in the section about vector line integrals. This article makes the opposite component (the i component) negative. • Why only -dx j_ ? Why dy_i doesn't get a minus sign, where normal vector = dy i_ - dx _j? • A couple things: Transforming dxi + dyj into dyi - dxj seems very much like taking a determinant. What's the relation? And two, couldn't you find a unit normal vector by finding the unit tangent vector, then making a vector perpendicular to it? i.e., using dot product to find perpendicular vector, or using a different vector and subtracting its projection onto the previous one? • So this is what I interpreted at the end ... The vector integral along the curve is equal to the area integral of divergence of the vector for the area formed by the curve … it would seem like the divergence would account for more stuff leaving because it’s the whole area while the line integral accounts for only what is happening along the curve? • Does anyone know what this symbol means?

ψ

I found it in an equation that ran like so: a ψ b = a^2 - 2b. Any help would be appreciated, as I've asked everyone I know and no one seems to have any idea. Thanks.
(1 vote) • That sounds like a question that you see on the SAT every now and then. They just define a new operation (along the lines of +, *, ^, etc.) and tell you what it does when you perform the operation with two given numbers. In other words, this symbol used for this purpose is unique to this problem and is not something where you could ask a random math professor at Princeton or something and they would tell you, "Oh yeah, that means a^2 - 2b."

It is the Greek letter psi, though. It makes a sound of "ps," and is the first letter of, for example, the Greek word from which the English word "psychology" is derived.