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Chapter 11: Q. 6 (page 900)

Giveprecisemathematicaldefinitionsordescriptionsofeachoftheconceptsthatfollow.Thenillustratethedefinitionwithagraphoranalgebraicexample.theantiderivativeofavectorfunctionr(t)

Short Answer

Expert verified

鈭�r(t)dt=鈭�(x(t)i+y(t)j+z(t)k)dt=i鈭�x(t)dt+j鈭�y(t)dt+k鈭�z(t)dt

Step by step solution

01

Step 1. Given information is:

A vector function r(t)

02

Step 2. Antiderivative of r(t)

Letx=x(t),y=y(t),andz=z(t)bethreereal-valuedfunctionswithantiderivatives.鈭�x(t)dt,鈭�y(t)dt,and鈭�z(t)dt,respectively,onsomeintervalI鈯�R.Thenthevectorfunction鈭�r(t)dt=鈭�(x(t)i+y(t)j+z(t)k)dt=i鈭�x(t)dt+j鈭�y(t)dt+k鈭�z(t)dtisanantiderivativeofthevectorfunctionr(t)=x(t),y(t),z(t).

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Most popular questions from this chapter

Given a vector-valued function r(t) with domain 鈩�, what is the relationship between the graph of r(t) and the graph of kr(t), where k > 1 is a scalar?

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role="math" localid="1649566464308" x鈥�(t)=1+x2,y鈥�(t)=x2,x(0)=0,y(0)=1. ( HINT: Start by solving the initial-value problemrole="math" localid="1649566360577" x鈥�(t)=1+x2,x(0)=0.)

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(1,3t,t3)+(t,t2,t3)

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r(t)=sint,sintcost,cos2t

Let r1(t)and r2(t) both be differentiable three-component vector functions. Prove that

ddtr1(t)脳r2(t)=r1'(t)脳r2(t)+r1(t)脳r2'(t) (This is Theorem 11.11 (d).)
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