Part 1/5:

Understanding the Derivative of a Vector Function

The concept of calculating the derivative of a vector function is fundamental in vector calculus. This process can be straightforwardly understood by examining the derivative component-wise. The theorem and its proof provide a clear framework for this, illustrating that the derivative of a vector function is simply the vector composed of the derivatives of its individual components.

Theorem: Derivative of a Vector Function

Given a vector function (\mathbf{r}(t)) which has components defined by differentiable functions (f(t)), (g(t)), and (h(t)), the derivative is obtained by differentiating each component separately.

Mathematically, if:

[

\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}

]