Part 4/5:
This derivation confirms that the derivative of the vector function is obtained by differentiating each component separately.
Significance and Applications
This theorem simplifies the process of differentiating vector functions in multiple fields such as physics, engineering, and computer graphics. For example:
Kinematics: Derivatives of position vectors give velocity and acceleration.
Electromagnetism: Differentiating vector fields to compute electric and magnetic field variations.
Motion Planning: Derivatives of paths in space for velocity and acceleration profiles.
Summary
- The derivative of a vector function with respect to (t) can be calculated by differentiating each component function.