PEPCVF - Derivatives of Vector-Valued Functions

Derivatives of Vector-Valued Functions

Differentiation of Vector-Valued Functions

The derivative of a vector function is defined in much the same way as real-valued functions: $\vec{r}\,'\left(t\right)=\frac{d\vec{r}\,}{dt}=\lim_{h \to 0}\frac{\vec{r}\,\left(t+h\right)-\vec{r}\,\left(t\right)}{h}$, if this limit exists. Visualization and derivation of the derivative of a position vector-valued function is the focus of the next presentation.

If r(t) = f(t) i + g(t) j + h(t) k, where f, g, and h are differentiable functions, then r '(t) = f '(t) i + g'(t) j+ h'(t) k = < f '(t), g'(t), h'(t)>. The derivative vector r'(t) may be visualized as a tangent vector to the curve defined by r at the point P, provided that r' (t) exists and r(t)≠ 0. The tangent line to the curve at P is the line through P parallel to the tangent vector r (t). The unit tangent vector is given by $\vec{T}\,\left(t\right)=\frac{\vec{r}\,\left(t\right)}{\left|\vec{r}\,\left(t\right)\right|}$.

Differentiation rules for real-valued functions have their counterparts for vector-valued functions as seen below.

Examples of vector-valued function derivatives and their geometric representations are illustrated below.

Velocity, Speed, Acceleration Along a Plane Curve

Image Caption: Isaac William Rowan Hamilton (1805-1865) was an Irish mathematician who spent many years developing a system of vector-like quantities called quarternions, although his work with quarternions never produced good models for physical phenomena. It was not until the latter half of the 19^th century when the Scottish physicist James Maxwell reformulated Hamilton's quarternions into a form useful in representing force, velocity, and acceleration.

As an object moves along a curve in the plane, the coordinates x and y are functions of time t and written as x = x(t) and y = y(t). Thus the position vector r(t) is written as r(t) = x(t) i + y(t) j. Just as with real-valued functions where the velocity is given by the first derivative, the velocity vector is also the tangent vector (first derivative of the position vector) and points in the direction of the tangent line.

$v\left(t\right)=r'\left(t\right)=x'\left(t\right)i+u'\left(t\right)i+y'\left(t\right)j=\frac{d\vec{r}\,}{dt}=\lim_{h \to 0}\frac{\vec{r}\,\left(t+h\right)-\vec{r}\,\left(t\right)}{h}$

The speed of the object at time t is the magnitude of the velocity vector, i.e., |v(t)| = |r '(t)| = $\frac{ds}{dt}=\sqrt[]{\left[x'\left(t\right)\right]^2+\left[y'\left(t\right)\right]^2}$ = rate of change of distance with respect to time. The presentation below focuses on interpreting what the derivative of a position vector looks like for two different parameterizations of the same curve.

Just as for real-valued functions, the second derivative of a vector function r is the derivative of r', i.e., r'' = (r ')'. The derivative a(t)= v '(t) = r''(t) = x''(t) i + y''(t)j is the acceleration vector of an object at time t. The next presentation illustrates how velocity, speed, and acceleration are found for curves in both the plane and in space.

An application of velocity and acceleration using unit vectors and involving projectile motion is the focus of the next two presentations.

Another example involving motion in the plane is provided below.

 

Derivatives of Vector-Valued Functions Practice

Derivatives of Vector-Valued Functions: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of derivatives of vector-valued functions.

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