Derivative of Cosine with unit vector

[Blue Kangaroo]

Homework Statement

Take ∂²E/∂t² E(r,t)=E₀cos((k(u^·r−ct)+φ) in which u^ is a unit vector.

Homework Equations

d/dx(cosx)=-sinx

The Attempt at a Solution

I had calc 3 four years ago and can't for the life of me remember how to differentiate the unit vector. I came up with -c²u²^cos(k(u^r-ct)+φ), but I feel like I'm not doing that right.

 

[RPinPA]

I think in this equation $\hat u$ is a constant in time. It's the direction of motion of this wave. And $r$ is held constant when doing the partial with respect to $t$. So you don't need to differentiate it, just treat $\hat u \cdot r$ as a constant, which it is.

 

[Blue Kangaroo]

I think in this equation $\hat u$ is a constant in time. It's the direction of motion of this wave. And $r$ is held constant when doing the partial with respect to $t$. So you don't need to differentiate it, just treat $\hat u \cdot r$ as a constant, which it is.

Ah I made a mistake. I meant to write ∂²E/∂r²

 

[RPinPA]

How is the unit vector defined? Unit vector in what direction?

 

[vela]

Ah I made a mistake. I meant to write ∂²E/∂r²

You're using the symbol r to stand for different things, so it's not clear what you're trying to do.

There's the vector $\vec r$, which is apparently what appears in the argument of the cosine. It wouldn't make sense to write $r\cdot \hat u$ since $r=|\vec r|$ isn't a vector. On the other hand, you say you're trying to calculate $\partial^2 E/\partial r^2$. Is $r$ here the magnitude of $\vec{r}$?