Vector Field / Line Integral

Calculus Level 3

The vector field F ⃗ \vec{F} has components ( F x , F y , F z ) = ( x y , y z , z x ) (F_x,F_y,F_z) = (x y, y z , z x) . Determine the line integral of the vector field over a straight-line path from ( 0 , 0 , 0 ) (0,0,0) to ( 1 , 2 , 3 ) (1,2,3) .

If the result can be expressed as a b \large{\frac{a}{b}} , where a a and b b are positive co-prime integers, what is a + b a + b ?

∫ C F ⃗ ⋅ d ℓ ⃗ = a b \large{\int_C \vec{F} \cdot \vec{d \ell} = \frac{a}{b}}


The answer is 26.

Steven Chase by Steven Chase , 37, USA

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2 solutions

Otto Bretscher
Nov 15, 2018

If we parameterize the given path as x = t , y = 2 t , z = 3 t x=t, y=2t, z=3t , with 0 ≤ t ≤ 1 0\leq t\leq 1 , then ∫ C x y d x + y z d y + z x d z = ∫ 0 1 ( 2 t 2 + 6 t 2 × 2 + 3 t 2 × 3 ) d t = 23 3 \int_C xydx+yzdy+zxdz=\int_{0}^{1}(2t^2+6t^2\times2+3t^2\times 3)dt={\frac{23}{3}} . The answer is 26 \boxed{26} .

Thank you for posting this! We need more vector calculus problems on Brilliant! (I will post one once in a while.)

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Indeed. Vector calculus problems are fun, and they are too few. I enjoyed yours as well

Steven Chase - 2 years, 6 months ago

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I'm teaching a vector calc class right now (grading a test today), and I might just post some of my exam problems.

Otto Bretscher - 2 years, 6 months ago

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Sir, I'll be waiting for those problems.......!!!

Aaghaz Mahajan - 2 years, 6 months ago

This is not different from Dr. Bretscher's. It is just expressed differently so as to show the path integral set up. r ( t$_$ ) :=Evaluate [ t ( { 1 , 2 , 3 } − { 0 , 0 , 0 } ) + { 0 , 0 , 0 } ] ⟹ { t , 2 t , 3 t } r(\text{t\$\_\$})\text{:=}\text{Evaluate}[t (\{1,2,3\}-\{0,0,0\})+\{0,0,0\}] \Longrightarrow \{t,2 t,3 t\} f ( { x$_$ , y$_$ , z$_$ } ) := { x y , y z , x z } f(\{\text{x\$\_\$},\text{y\$\_\$},\text{z\$\_\$}\})\text{:=}\{x y,y z,x z\} ∫ 0 1 f ( r ( t ) ) . ∂ r ( t ) ∂ t d t ⟹ 23 3 \int_0^1 f(r(t)).\frac{\partial r(t)}{\partial t} \, dt \Longrightarrow \frac{23}{3}

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