Given that: M = îo – Ôtane 1-Transform this vector into Cartesian coordinates? 2- Find the divergence of this vector in Cartesian coordinates?
Given that: M = îo – Ôtane 1-Transform this vector into Cartesian coordinates? 2- Find the divergence of this vector in Cartesian coordinates?
Related questions
Question
![### Question
Given that:
\[ \vec{M} = \hat{r_0} - \hat{\theta} \tan{\theta} \]
1. Transform this vector into Cartesian coordinates.
2. Find the divergence of this vector in Cartesian coordinates.
### Explanations
1. **Transforming Vector into Cartesian Coordinates**:
To transform \( \vec{M} \) from polar coordinates (\( \hat{r_0} \) and \( \hat{\theta} \)) to Cartesian coordinates (\( \hat{i} \) and \( \hat{j} \)), we need to use the relationships between polar and Cartesian unit vectors:
- \( \hat{r_0} = \cos{\theta} \hat{i} + \sin{\theta} \hat{j} \)
- \( \hat{\theta} = -\sin{\theta} \hat{i} + \cos{\theta} \hat{j} \)
Substitute these into the given vector equation:
\[ \vec{M} = (\cos{\theta} \hat{i} + \sin{\theta} \hat{j}) - \tan{\theta} (-\sin{\theta} \hat{i} + \cos{\theta} \hat{j}) \]
Simplify this expression to get \( \vec{M} \) in terms of \( \hat{i} \) and \( \hat{j} \).
2. **Finding the Divergence**:
Once \( \vec{M} \) is expressed in Cartesian coordinates, apply the divergence operator, \( \nabla \cdot \vec{M} \), which in Cartesian coordinates is:
\[ \nabla \cdot \vec{M} = \frac{\partial M_x}{\partial x} + \frac{\partial M_y}{\partial y} \]
where \( M_x \) and \( M_y \) are the components of \( \vec{M} \) in the \( \hat{i} \) and \( \hat{j} \) directions respectively.
These steps will help in transforming the vector and then finding its divergence in Cartesian coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94afdee9-ee55-4a62-88bf-4de838a1ae93%2F28b8b8dc-3ecc-4775-b6fc-0bfa6c686cb4%2F2hl0brs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question
Given that:
\[ \vec{M} = \hat{r_0} - \hat{\theta} \tan{\theta} \]
1. Transform this vector into Cartesian coordinates.
2. Find the divergence of this vector in Cartesian coordinates.
### Explanations
1. **Transforming Vector into Cartesian Coordinates**:
To transform \( \vec{M} \) from polar coordinates (\( \hat{r_0} \) and \( \hat{\theta} \)) to Cartesian coordinates (\( \hat{i} \) and \( \hat{j} \)), we need to use the relationships between polar and Cartesian unit vectors:
- \( \hat{r_0} = \cos{\theta} \hat{i} + \sin{\theta} \hat{j} \)
- \( \hat{\theta} = -\sin{\theta} \hat{i} + \cos{\theta} \hat{j} \)
Substitute these into the given vector equation:
\[ \vec{M} = (\cos{\theta} \hat{i} + \sin{\theta} \hat{j}) - \tan{\theta} (-\sin{\theta} \hat{i} + \cos{\theta} \hat{j}) \]
Simplify this expression to get \( \vec{M} \) in terms of \( \hat{i} \) and \( \hat{j} \).
2. **Finding the Divergence**:
Once \( \vec{M} \) is expressed in Cartesian coordinates, apply the divergence operator, \( \nabla \cdot \vec{M} \), which in Cartesian coordinates is:
\[ \nabla \cdot \vec{M} = \frac{\partial M_x}{\partial x} + \frac{\partial M_y}{\partial y} \]
where \( M_x \) and \( M_y \) are the components of \( \vec{M} \) in the \( \hat{i} \) and \( \hat{j} \) directions respectively.
These steps will help in transforming the vector and then finding its divergence in Cartesian coordinates.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
