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Mathematics LibreTexts

4.1: Differentiation and Integration of Vector Valued Functions

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The formal definition of the derivative of a vector valued function is very similar to the definition of the derivative of a real valued function.

Definition: The Derivative of a Vector Valued Function

Let r(t) be a vector valued function, then

r(t)=limh0r(t+h)r(t)h.

Because the derivative of a sum is the sum of the derivative, we can find the derivative of each of the components of the vector valued function to find its derivative.

Example 4.1.1

ddt(3ˆi+sintˆj)=costˆj

ddt(3t2ˆi+cos(4t)ˆj+tetˆk)=6tˆi4sin(t)ˆj+(et+tet)ˆk

Properties of Vector Valued Functions

All of the properties of differentiation still hold for vector values functions. Moreover because there are a variety of ways of defining multiplication, there is an abundance of product rules.

Suppose that v(t) and w(t) are vector valued functions, f(t) is a scalar function, and c is a real number then

  1. ddt(v(t)+w(t))=ddtv(t)+ddtw(t),
  2. ddtcv(t)=cddtv(t),
  3. ddt(f(t)v(t))=f(t)v(t)+f(t)v(t),
  4. (v(t)w(t))=v(t)w(t)+v(t)w(t),
  5. (v(t)×w(t))=v(t)×w(t)+v(t)×w(t),
  6. ddtv(f(t))=v(t)(f(t))f(t).
Example 4.1.2

Show that if r is a differentiable vector valued function with constant magnitude, then

rr=0.

Solution

Since r has constant magnitude, call its magnitude k,

k2=|r|2=rr.

Taking derivatives of the left and right sides gives

0=(rr)=rr+rr

=rr+rr=2rr.

Divide by two and the result follows

Integration of vector valued functions

We define the integral of a vector valued function as the integral of each component. This definition holds for both definite and indefinite integrals.

Example 4.1.3

Evaluate

(sint)ˆi+2tˆj8t3ˆkdt.

Solution

Just take the integral of each component

(sint)dtˆi+2tdtˆj8t3dtˆk.

=(cost+c1)ˆi+(t2+c2)ˆj+(2t4+c3)ˆk.

Notice that we have introduce three different constants, one for each component.

Larry Green (Lake Tahoe Community College)

  • Integrated by Justin Marshall.


4.1: Differentiation and Integration of Vector Valued Functions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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