Vector Calc 8-3-2020

Calculus Level 3

Consider the vector field:

F ⃗ = ( F x , F y ) = ( 2 x y + 1 , x 2 + 1 ) \vec{F} = (F_x, F_y) = (2 x y + 1, x^2 + 1)

Evaluate the line integral of the vector field over the path from ( x , y ) = ( − 1 , 1 ) (x,y) = (-1,1) to ( x , y ) = ( 1 , 1 ) (x,y) = (1,1) over the curve y = x 2 y = x^2 .

∫ F ⃗ ⋅ d ℓ ⃗ \int \vec{F} \cdot \vec{d \ell}


The answer is 2.0.

Steven Chase by Steven Chase , 37, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Hosam Hajjir
Aug 3, 2020

By inspection of the vector field F ⃗ \vec{F} , it turns out to be the gradient of a function ϕ ( x , y ) \phi(x, y) . It is straight forward to see that ϕ ( x , y ) = x 2 y + x + y \phi(x, y) = x^2 y + x + y , hence,

∫ C F ⃗ ⋅ d l ⃗ = ϕ ( 1 , 1 ) − ϕ ( − 1 , 1 ) = ( 1 + 1 + 1 ) − ( 1 − 1 + 1 ) = 3 − 1 = 2 \large \displaystyle \int_C \vec{F} \cdot \vec{dl} = \phi(1, 1) - \phi(-1, 1) = (1 + 1 + 1) - (1 - 1 + 1) = 3 - 1 = \boxed{2}

3 Helpful 3 Interesting 4 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...