Mastering Vector Derivatives: Differentiating Dot And Cross Products

What is the derivative of (u•v)

(u’•v)+(u•v’)

The derivative of (u・v), where u and v are vectors, depends on whether we are taking the dot product or the cross product.

If we are taking the dot product, the formula for the derivative is:

d/dt (u・v) = (du/dt)・v + u・(dv/dt)

Here, the dot (・) represents the dot product, and du/dt and dv/dt represent the derivatives of vectors u and v, respectively, with respect to time t.

We can see that the derivative of the dot product is a vector itself, and it consists of two components: (i) the derivative of u with respect to time multiplied by vector v, and (ii) vector u multiplied by the derivative of v with respect to time.

On the other hand, if we are taking the cross product, then there is no derivative of (u・v), since the cross product is not a differentiable operation.

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