How Do You Prove a Vector is Unit Along a Parametric Curve?

In summary, a directional derivative is a measure of how a function changes in a specific direction, calculated using the dot product of the gradient and direction vectors. It is significant in understanding the behavior of a function and has various applications. The proof of directional derivative is derived from the definition of a derivative in multivariable calculus. It can be negative depending on the direction of the vector.
  • #1
brunette15
58
0
Hey everyone,

I am given the following function f(x,y) = xy+x+y along the curve x(s)=rcos(s/r) and y(s)=rsin(s/r). I have to show that (dx/s)i + (dy/ds)j is a unit vector.

I am unsure where to begin with this :/

Can anyone please give me some hints/ideas on how to approach this question?
 
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  • #2
Hi brunette15,

What does it mean when a vector is a unit vector? Let's start by finding $\d{y}{s}$ and $\d{x}{s}$.
 

FAQ: How Do You Prove a Vector is Unit Along a Parametric Curve?

What is a directional derivative?

A directional derivative is a measure of how a function changes with respect to a specific direction. It allows us to calculate the rate of change of a function in a particular direction, rather than just in the x or y direction.

How is the directional derivative calculated?

The directional derivative is calculated using the dot product of the gradient of the function and the direction vector. This can be expressed as: Df(x,y) = ∇f(x,y) · u, where ∇f(x,y) is the gradient vector and u is the direction vector.

What is the significance of the directional derivative?

The directional derivative is important because it allows us to find the slope of a function in a particular direction. This can be useful in optimizing functions and understanding the behavior of a function in different directions. It also has applications in physics, engineering, and economics.

How is the proof of directional derivative derived?

The proof of directional derivative is derived using the definition of a derivative in multivariable calculus. It involves taking the limit of the difference quotient as the change in the independent variable approaches 0. This leads to the formula for the directional derivative mentioned in the second question.

Can the directional derivative be negative?

Yes, the directional derivative can be negative. It depends on the direction of the vector u. If the direction is opposite to the gradient vector, the directional derivative will be negative. This means that the function is decreasing in that direction.

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