Part 1/6:
Understanding Derivatives and Unit Tangent Vectors in Multivariable Calculus
In the study of vector calculus, especially when analyzing curves in space, understanding derivatives and tangent vectors is fundamental. Recently, an example was discussed that demonstrates the process of differentiating a vector function, calculating the unit tangent vector at specific points, and understanding their geometric significance.
The Vector Function ( r(t) )
The vector function under consideration is:
[
r(t) = (1 + t^3) \mathbf{i} + t e^{-t} \mathbf{j} + \sin(2t) \mathbf{k}
]
This function describes a curve in three-dimensional space, where each component varies with the parameter ( t ).