Differentiate time derivative w/ respect to generalized var.

• buildingblocs
In summary, the conversation discusses solving for ∂v/∂θ and ∂v/∂r in cylindrical coordinates, with attention to the unit vectors eθ and eρ. The solution provided in the attached image is incorrect as it solves for ∂v/∂θ and ∂v/∂r in terms of ∂v/∂θ̇ and ∂v/∂ṙ. The correct solution involves expressing the cylindrical unit vectors in terms of Cartesian unit vectors.
buildingblocs

Homework Statement

Solve ∂v/∂θ and ∂v/∂r. (refer to attached image for equations)

Homework Equations

Refer to attached image. note that the velocity is expressed in cylindrical coordinates and attention must be paid to the directional unit vectors eθ and eρ.[/B]

The Attempt at a Solution

Have solution (refer to attached image). However would like to understand the general steps involved such to apply them to other equations with generalised variables and time derivatives.

Attachments

• IMAG0206[1].jpg
31.4 KB · Views: 453
It looks like you took ##\frac{\partial v}{\partial \dot\theta}## and ##\frac{\partial v}{\partial \dot r}##, not ##\frac{\partial v}{\partial \theta}## and ##\frac{\partial v}{\partial r}##.

Can you express ##\hat e_\rho## and ##\hat e_\theta## in terms of ##\hat x## and ##\hat y##?

buildingblocs
1)Yes you are correct, I made that mistake.

2)Here are the cylindrical unit vectors expressed in Cartesian unit vectors.

C=Asinθ+Bcosθ
D=Acosθ-Bsinθ

Unit vectors, therefore A=B=1;

C=sinθ+cosθ
D=cosθ-sinθ

1. What is a time derivative?

A time derivative is the mathematical expression for the rate of change of a quantity with respect to time. It is denoted by the symbol "d/dt" and is used to calculate how fast a quantity is changing at a specific moment in time.

2. What is a generalized variable?

A generalized variable is a variable that represents a physical quantity in a system. It can take on different values depending on the context, and is often used in physics to describe systems with multiple variables.

3. How do you differentiate with respect to a generalized variable?

To differentiate with respect to a generalized variable, you need to apply the chain rule. This involves taking the derivative of the function with respect to the generalized variable and then multiplying it by the derivative of the generalized variable with respect to time.

4. What is the significance of differentiating with respect to a generalized variable?

Differentiating with respect to a generalized variable allows us to analyze how a system changes over time. It helps us understand the relationship between different quantities in a system and how they affect each other.

5. What are some common examples of using time derivatives with respect to generalized variables?

Some common examples include calculating the velocity of an object at a specific time, determining the rate of change of a chemical concentration in a reaction, and finding the acceleration of a moving particle in a system with multiple variables.

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