Section 3.2 - Derivatives and Integrals of Vector Functions

Section 3.2 - Derivatives and Integrals of Vector Functions

Derivatives of Vector Functions

In calculus, the derivative of a vector function describes the rate of change of the vector with respect to a variable, typically time or position. This concept extends the idea of differentiation from scalar functions to functions whose outputs are vectors.

• Definition: The derivative r'(t) of a vector function r(t) is defined as: $\frac{d\vec{r}}{dt} = \lim_{h \to 0} \frac{\vec{r}(t + h) - \vec{r}(t)}{h}$ if this limit exists.

• Tangent Vector: For points P and Q on a curve, the vector r'(t) at P is tangent to the curve at P, provided that P moves along the curve as t varies. The unit tangent vector at point P is: $\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$

• Component-wise Differentiation: If $\vec{r}(t) = f(t)\vec{i} + g(t)\vec{j} + h(t)\vec{k}$, where f, g, h are differentiable functions, then: $\vec{r}'(t) = f'(t)\vec{i} + g'(t)\vec{j} + h'(t)\vec{k}$

Example 1: Find the derivative of $\vec{r}(t) = (1 + t^2)\vec{i} + te^{t^2}\vec{j} + \sin(t^2)\vec{k}$ and its unit tangent vector at $t = 0$.

• Compute derivatives of each component:

  • $\frac{d}{dt}(1 + t^2) = 2t$

  • $\frac{d}{dt}(te^{t^2}) = e^{t^2} + 2t^2e^{t^2}$

  • $\frac{d}{dt}(\sin(t^2)) = 2t\cos(t^2)$

• At $t = 0$, substitute to find $\vec{r}'(0)$ and then normalize to get the unit tangent vector.

Example 2: For the curve $\vec{r}(t) = t\vec{i} + (2 - t)\vec{j}$, find $\vec{r}'(t)$ and sketch the position vector $\vec{r}(1)$ and the tangent vector $\vec{r}'(1)$.

• $\vec{r}'(t) = \vec{i} - \vec{j}$

• At $t = 1$, $\vec{r}(1) = \vec{i} + \vec{j}$, $\vec{r}'(1) = \vec{i} - \vec{j}$

Example 3: Show that if $|\vec{r}(t)| = c$ (a constant), then $\vec{r}'(t)$ is orthogonal to $\vec{r}(t)$ for all t.

• Differentiate $|\vec{r}(t)|^2 = c^2$ with respect to t: $\frac{d}{dt}(\vec{r}(t) \cdot \vec{r}(t)) = 2\vec{r}(t) \cdot \vec{r}'(t) = 0$ Thus, $\vec{r}(t) \cdot \vec{r}'(t) = 0$, so they are orthogonal.

Properties of Derivatives of Vector Functions

• $\frac{d}{dt}(c\vec{r}(t)) = c\frac{d\vec{r}(t)}{dt}$, where c is a constant.

• $\frac{d}{dt}(\vec{r}_1(t) + \vec{r}_2(t)) = \vec{r}_1'(t) + \vec{r}_2'(t)$

• $\frac{d}{dt}(f(t)\vec{r}(t)) = f'(t)\vec{r}(t) + f(t)\vec{r}'(t)$

• $\frac{d}{dt}(\vec{r}_1(t) \cdot \vec{r}_2(t)) = \vec{r}_1'(t) \cdot \vec{r}_2(t) + \vec{r}_1(t) \cdot \vec{r}_2'(t)$

• $\frac{d}{dt}(\vec{r}_1(t) \times \vec{r}_2(t)) = \vec{r}_1'(t) \times \vec{r}_2(t) + \vec{r}_1(t) \times \vec{r}_2'(t)$

• $\frac{d}{dt}(f(\vec{r}(t))) = f'(\vec{r}(t))\vec{r}'(t)$

Integrals of Vector Functions

Integration of vector functions extends the concept of integration to functions whose outputs are vectors. The definite and indefinite integrals are defined component-wise.

• Definite Integral: For a continuous vector function $\vec{r}(t)$: $\int_a^b \vec{r}(t) dt = \int_a^b f(t)dt\vec{i} + \int_a^b g(t)dt\vec{j} + \int_a^b h(t)dt\vec{k}$

• Fundamental Theorem of Calculus (Vector Version): $\int_a^b \vec{r}'(t) dt = \vec{r}(b) - \vec{r}(a)$

• Indefinite Integral: The indefinite integral of $\vec{r}(t)$ is defined as: $\int \vec{r}(t) dt = \vec{R}(t) + \vec{C}$ where $\vec{R}(t)$ is any antiderivative of $\vec{r}(t)$ and $\vec{C}$ is a constant vector.

Example 4: If $\vec{r}(t) = 2\cos(t^2)\vec{i} + \sin(t^2)\vec{j} + 2t\vec{k}$, find $\int \vec{r}(t) dt$.

• Integrate each component separately:

  • $\int 2\cos(t^2) dt$ (requires substitution or special functions)

  • $\int \sin(t^2) dt$ (requires substitution or special functions)

  • $\int 2t dt = t^2$

Summary Table: Properties of Derivatives and Integrals of Vector Functions

Operation	Formula	Description
Derivative	$\frac{d\vec{r}}{dt}$	Rate of change of vector function
Unit Tangent Vector	$\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$	Direction of motion along curve
Definite Integral	$\int_a^b \vec{r}(t) dt$	Net change in vector function over [a, b]
Fundamental Theorem	$\int_a^b \vec{r}'(t) dt = \vec{r}(b) - \vec{r}(a)$	Relates derivative and integral

Additional info: The notes also briefly mention the extension of the Fundamental Theorem of Calculus to vector functions and provide examples of differentiation and integration of vector functions, including the use of component-wise operations.