Evaluate the Following Line Integral Part 2

[bugatti79]

Homework Statement

$\displaystyle \int_c zdx+xdy+ydz$ where C is given by $t^2\vec i +t^3 \vec j +t^2 \vec k$

Can this $\displaystyle \int_c zdx+xdy+ydz$ be written as $\displaystyle \int_c z\vec i+x \vec j+y \vec k$?

I believe I need to evalute the integral $\displaystyle \int_c \vec F( \vec r (t)) d\vec r(t)$...?

How do I proceed?

 

[I like Serena]

Hi again bugatti79!

Homework Statement

$\displaystyle \int_c zdx+xdy+ydz$ where C is given by $t^2\vec i +t^3 \vec j +t^2 \vec k$

Can this $\displaystyle \int_c zdx+xdy+ydz$ be written as $\displaystyle \int_c z\vec i+x \vec j+y \vec k$?

It should be:
$\int_c (z\vec i+x \vec j+y \vec k) \cdot d\vec r(t)$

I believe I need to evalute the integral $\displaystyle \int_c \vec F( \vec r (t)) d\vec r(t)$...?

How do I proceed?

What would $d\vec r(t)$ be?

 

[bugatti79]

Hi again bugatti79!



It should be:
$\int_c (z\vec i+x \vec j+y \vec k) \cdot d\vec r(t)$

How did you establish this?

What would $d\vec r(t)$ be

From what you have shown I rearranged to find $d\vec r = \vec i dx+\vec j dy+ \vec k dz$...?

No, that wouldn't be right as dr is a function of t...

$d\vec r = d(t^2 \vec i +t^3\vec j+ t^2\vec k)$

 

[I like Serena]

How did you establish this?

What you have is a dot product:
$zdx+xdy+ydz=(z,x,y) \cdot (dx,dy,dz)$

The vector (z,x,y) corresponds to your function $\vec F(\vec r)$

The vector (dx,dy,dz) corresponds to your $d\vec r(t)$

The vector (x,y,z) corresponds to your $\vec r(t)$

From what you have shown I rearranged to find $d\vec r = \vec i dx+\vec j dy+ \vec k dz$...?

No, that wouldn't be right as dr is a function of t...

$d\vec r = d(t^2 \vec i +t^3\vec j+ t^2\vec k)$

Yes, the latter.
Can you work this out as a derivative?

 

[bugatti79]

Hi again bugatti79!



It should be:
$\int_c (z\vec i+x \vec j+y \vec k) \cdot d\vec r(t)$




What would $d\vec r(t)$ be?

$\displaystyle \int_c (z\vec i+x \vec j+y \vec k) \cdot d\vec r(t)=\int_c (t^2 \vec i+t^2 \vec j+t^3 \vec k) \cdot (2t \vec i + 3t^2 \vec j + 2t \vec k)dt = \int_{0}^{1} (2t^3 +3t^4+2t^4) dt$...?

 

[I like Serena]

$\displaystyle \int_c (z\vec i+x \vec j+y \vec k) \cdot d\vec r(t)=\int_c (t^2 \vec i+t^2 \vec j+t^3 \vec k) \cdot (2t \vec i + 3t^2 \vec j + 2t \vec k)dt = \int_{0}^{1} (2t^3 +3t^4+2t^4) dt$...?

Yep!
(That was easy, was it not? )

 

[bugatti79]

Cheers! There will be more trickier ones along the way. :-) Thanks