Problem 196 - Calculus 3

Question

The volume of a right circular cylinder is given by \(V(r, h)=\pi r^{2} h .\) Find the differential \(d V .\) Interpret the formula geometrically.

Step-by-Step Solution

Answer

The differential is \( dV = 2\pi rh \, dr + \pi r^2 \, dh \). It shows how volume changes with small changes in radius and height.

Step 1: Write the Formula for Volume

The volume of a right circular cylinder is given by the formula \( V(r, h) = \pi r^2 h \). This formula expresses the volume in terms of its radius \( r \) and height \( h \).

Step 2: Differentiate with Respect to Variables

To find the differential \( dV \), we need to take the partial derivatives of \( V \) with respect to \( r \) and \( h \). According to the formula, use the partial derivatives: \( \frac{\partial V}{\partial r} = 2\pi rh \) and \( \frac{\partial V}{\partial h} = \pi r^2 \).

Step 3: Apply the Total Differential Formula

The total differential \( dV \) is given by the formula: \( dV = \frac{\partial V}{\partial r} \, dr + \frac{\partial V}{\partial h} \, dh \). Substitute the partial derivatives: \( dV = 2\pi rh \, dr + \pi r^2 \, dh \).

Step 4: Interpret Geometrically

Geometrically, the differential \( dV \) represents the approximate change in volume of the cylinder when the radius changes by \( dr \) and the height changes by \( dh \). The term \( 2\pi rh \, dr \) represents the change in volume due to a small change in radius, creating a thin cylindrical shell, and \( \pi r^2 \, dh \) represents the change in volume due to a small change in height, creating a thin disc.

Key Concepts

Partial Derivatives

When dealing with functions that have multiple variables, like the volume function of a cylinder, partial derivatives are a powerful tool. A partial derivative tells us how the function changes if we vary one variable at a time, while keeping other variables constant.

• For our cylinder, the function is given as \( V(r, h) = \pi r^2 h \). This means the volume \( V \) is a function of both the radius \( r \) and the height \( h \).
• The partial derivative of \( V \) with respect to \( r \), written as \( \frac{\partial V}{\partial r} \), shows how the volume changes if only the radius changes. In this case, \( \frac{\partial V}{\partial r} = 2\pi rh \).
• Similarly, the partial derivative of \( V \) with respect to \( h \) is \( \frac{\partial V}{\partial h} = \pi r^2 \), which shows how the volume changes if only the height changes.

By understanding and calculating these partial derivatives, we can analyze the sensitivity of the volume to changes in each dimension separately.

Total Differential

The total differential is a mathematical concept that provides an approximate change in a function due to small changes in each of its variables. For a function like the volume of a cylinder, the total differential can be seen as a combination of the effects of changing each variable one at a time.

• The total differential \( dV \) of the volume function \( V(r, h) = \pi r^2 h \) is given by \( dV = \frac{\partial V}{\partial r} \, dr + \frac{\partial V}{\partial h} \, dh \).
• For our cylinder, substituting the partial derivatives, we get \( dV = 2\pi rh \, dr + \pi r^2 \, dh \).
• This equation represents the overall change in the cylinder's volume. It aggregates small changes in radius and height, representing them as small linear increments from each variable.

The total differential is a valuable tool in calculus as it simplifies how we study small changes in multivariable functions and predicts their behavior in a local environment.

Geometry of Cylinders

Cylinders are a common 3D shape, and understanding their geometry helps when interpreting differentials geometrically. When considering changes in the cylinder's dimensions:

• The differential \( dV \) has a geometric interpretation. It reflects how small changes in radius and height produce volume changes.
• The term \( 2\pi rh \, dr \) represents the volume of a thin "shell" formed by increasing the radius slightly. Imagine peeling a thin layer from the surface of the cylinder.
• On the other hand, \( \pi r^2 \, dh \) describes the volume of a thin "disk" added to the height. This can be visualized as stacking another thin slice on top of the cylinder.
• Together, these small volume increments contribute to the differential expansion of the cylinder.

By focusing on these aspects, students can better visualize how tiny alterations in dimensions lead to changes in the cylinder's overall volume, reinforcing the connection between calculus and geometry.