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