# Integrating unit vector ρ

• I
• ys98
In summary, when integrating with cylindrical coordinates, it is necessary to express the variable being integrated in terms of the cylindrical coordinates. The second method presented, which uses the cylindrical coordinates ##\hat{\rho} = sin(\phi)\hat{i} + cos(\phi)\hat{j}##, is the correct approach. It is not possible to use cylindrical coordinates and still obtain the same solution without converting back to Cartesian coordinates.

#### ys98

When doing integration such as $\int_{0}^{2\pi} \hat{\rho} d\phi$ which would give us $2\pi \hat{\rho}$, must we decompose $\hat{ρ}$ into $sin(\phi) \hat{i} + cos(\phi) \hat{j}$ , then $\int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi$ , which would give us 0 instead?

Thanks

ys98 said:
When doing integration such as $\int_{0}^{2\pi} \hat{\rho} d\phi$ which would give us $2\pi \hat{\rho}$, must we decompose $\hat{ρ}$ into $sin(\phi) \hat{i} + cos(\phi) \hat{j}$ , then $\int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi$ , which would give us 0 instead?

Thanks

Which method looks correct to you?

PeroK said:
Which method looks correct to you?
The second one. My question is, is there anyway that I can keep using cylindrical coordinate without changing back to cartesian coordinate and get the same solution?

ys98 said:
The second one. My question is, is there anyway that I can keep using cylindrical coordinate without changing back to cartesian coordinate and get the same solution?

You must express ##\hat{\rho}## as a function of the variable with which you are integrating. Your second method looks the only viable option to me. That was using cylindrical coordinates. Using Cartesian coordinates would entail expressing the integral in terms of the Cartesian variables ##x## and ##y##.

ys98

## What is a unit vector ρ and why is it important in integration?

A unit vector ρ is a vector with a magnitude of 1 and is used to represent direction in vector calculus. It is important in integration because it allows for the calculation of integrals in multiple dimensions.

## How is unit vector ρ integrated in vector calculus?

In vector calculus, unit vector ρ is integrated using the dot product, which involves multiplying the components of the vector by the corresponding components of the integrand.

## What are the benefits of using unit vector ρ in integration?

Using unit vector ρ in integration allows for the simplification of calculations and the ability to integrate in multiple dimensions, making it a powerful tool in vector calculus.

## Can unit vector ρ be integrated in both polar and Cartesian coordinates?

Yes, unit vector ρ can be integrated in both polar and Cartesian coordinates, as it represents direction and can be used to calculate integrals in any coordinate system.

## Are there any common mistakes when integrating unit vector ρ?

One common mistake when integrating unit vector ρ is forgetting to apply the dot product when multiplying the vector components with the integrand. Another mistake is using the wrong coordinate system, which can result in incorrect calculations.