Time Derivatives of Unit Vectors for Polar Coordinates (Chain Rule) || 2D Coordinate Systems

Finding the Time Derivative of r-hat and theta-hat

In this section, we will focus on finding the time derivative of r-hat and theta-hat in polar coordinates. We will explore how these unit vectors change with respect to the angle theta and understand their relationship to Cartesian coordinates.

Differentiating r-hat and theta-hat with respect to theta

• The unit vectors r-hat and theta-hat in polar coordinates depend on the angle theta.

• Unlike i-hat and j-hat in Cartesian coordinates, r-hat and theta-hat change depending on the angle.

• The direction of the unit vectors is determined by the angle theta, which is determined by the position of the origin relative to our location.

Theta derivatives and relationships

• The derivative of r-hat with respect to theta is equal to theta hat (tangent unit vector).

• The derivative of theta hat with respect to theta is equal to minus r hat (negative radial unit vector).

• These relationships were derived by expressing the unit vectors in terms of i-hat and j-hat, differentiating their horizontal and vertical components, and finding these relationships.

Time derivatives using chain rule

• To find the time derivatives of r-hat and theta-hat, we apply the chain rule for differentiation.

• The time derivative of r hat is equal to theta dot times theta hat.

• The time derivative of theta hat is equal to minus theta dot times r hat.

• Theta dot represents the rate at which the angle changes with respect to time.

Time Derivatives of Vector Quantities

In this section, we will use the chain rule again to find the time derivatives of actual vector quantities using previously derived relationships.

Time derivative of r-hat

• The time derivative of r hat is found using the chain rule: d/dt(r hat) = (dr hat/dtheta) * (dtheta/dt).

• From the previous relationships, dr hat/dtheta is equal to theta hat.

• Therefore, the time derivative of r hat is equal to theta dot times theta hat.

Time derivative of theta-hat

• The time derivative of theta hat is found using the chain rule: d/dt(theta hat) = (dtheta hat/dtheta) * (dtheta/dt).

• From the previous relationships, dtheta hat/dtheta is equal to minus r hat.

• Therefore, the time derivative of theta hat is equal to minus theta dot times r hat.

Dependence on angle and time

• Both r-hat and theta-hat depend on the angle theta, which can also depend on time.

• If there is no change in angle with respect to time (theta dot = 0), both derivatives will be zero.

• The relationship between these unit vectors carries over into their time derivatives.

Summary

In this transcript, we focused on finding the time derivatives of r-hat and theta-hat in polar coordinates. We explored how these unit vectors change with respect to the angle theta and derived their relationships. Using the chain rule, we found that the time derivative of r-hat is equal to theta dot times theta-hat, while the time derivative of theta-hat is equal to minus theta dot times r-hat. These derivatives depend on both the angle and its rate of change with respect to time.