How to differentiate a vector with respect to time?

Question

Say I have: $\vec{A}(\vec{x}(t))$ and I want to find $\frac{d\vec{A}}{dt}$. Can someone please tell me how to do this? I am getting confused regarding the chain rule here because we can write out $\vec{A} = <A_x,A_y,A_z>$ and each of those is a function of time.

Answer 1

If

$$\mathbf{A} = A(\mathbf{x}(t))$$

Then

$$\frac{\text{d}\mathbf{A}}{\text{d}t} = \frac{\text{d}A}{\text{d}x}\frac{\text{d}x}{\text{d}t}$$

Now just extend the rule with the chain method for more than the single $x$ component...

Answer 2

Let's consider the case of the vector $\vec{A}$, you mentioned in the question.

$\vec{A}\ =\ A_x \hat{x}\ +\ A_y\hat{y}\ +\ A_z\hat{z}$

$\hat{x}$, $\hat{y}$ and $\hat{z}$ are the (fixed) basis vectors in 3-dimensional space. So, the operator $\frac{d}{dt}$ will act only on the components of $\vec{A}$.

$\frac{d\vec{A}}{dt}\ =\ \frac{dA_x}{dt}\hat{x}\ +\ \frac{dA_y}{dt}\hat{y}\ +\ \frac{dA_z}{dt}\hat{z}$

Answer 3

You can just do the time derivative of each component separately and then but them back together into the vector. If the time derivative of the components requires the chain rule, then so be it; you can still do each component by itself.