Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(w=x^{2} y+z^{2}, x=\rho \cos \theta \sin \phi, y=\rho \sin \theta \sin \phi\), and \(z=\rho \cos \phi\), find $$ \left.\frac{\partial w}{\partial \theta}\right|_{\rho=2, \theta=\pi, \phi=\pi / 2} $$

Step-by-Step Solution

Answer

The partial derivative is \( -1 \).

Step 1: Understand the Problem

We need to compute the partial derivative \( \frac{\partial w}{\partial \theta} \) using the given relationships for \( x, y, \) and \( z \). The formula for \( w \) is \( w = x^2 y + z^2 \), and \( x, y, \) and \( z \) are given in terms of spherical coordinates \( \rho, \theta, \) and \( \phi \). Once we get this expression, we will substitute the values \( \rho=2, \theta=\pi, \phi=\frac{\pi}{2} \).

Step 2: Substitute Spherical Coordinates

Substitute the given expressions \( x = \rho \cos \theta \sin \phi \), \( y = \rho \sin \theta \sin \phi \), and \( z = \rho \cos \phi \) into the equation \( w = x^2 y + z^2 \). This yields \[ w = (\rho \cos \theta \sin \phi)^2 (\rho \sin \theta \sin \phi) + (\rho \cos \phi)^2. \]

Step 3: Simplify the Expression for w

Expand and simplify the expression for \( w \): \[ w = \rho^3 (\cos^2 \theta)(\sin^3 \phi)(\sin \theta) + \rho^2 (\cos^2 \phi). \] This can be further simplified if needed, but it remains in terms of \( \rho, \theta, \phi \) for now.

Step 4: Differentiate with Respect to \(\theta\)

Differentiate the expression for \( w \) with respect to \( \theta \). Focus on the derivative of each term: \( (\cos^2 \theta)(\sin \theta) \). Use the product rule and chain rule to obtain: \[ \frac{\partial}{\partial \theta}(\cos^2 \theta)(\sin \theta) = (2\cos \theta)(-\sin \theta)(\sin \theta) + (\cos^2 \theta)(\cos \theta). \] This simplifies to: \[ -2\cos \theta \sin^2 \theta + \cos^3 \theta. \] The term \( \cos^2 \phi \) does not depend on \( \theta \) so its derivative is zero.

Step 5: Evaluate at Given Point

Substitute the given values \( \rho=2, \theta=\pi, \phi=\frac{\pi}{2} \) into the derivative expression. At these values, \( \cos \pi = -1 \), \( \sin \pi = 0 \), and \( \sin \frac{\pi}{2} = 1 \). Substitute these values into the expression from Step 4: \[ -2(-1)(0)^2 + (-1)^3 = 0 - 1 = -1. \]

Step 6: Conclusion

The partial derivative of \( w \) with respect to \( \theta \), evaluated at the given point, is \( -1 \). Therefore, \( \left.\frac{\partial w}{\partial \theta}\right|_{\rho=2, \theta=\pi, \phi=\pi/2} = -1 \).

Key Concepts

Spherical Coordinates

Spherical coordinates are a system for representing points in three-dimensional space using three numbers. These are usually denoted by \( \rho \), \( \theta \), and \( \phi \). Each of these numbers has a specific geometric interpretation:

• \( \rho \) represents the radial distance from the origin to the point.
• \( \theta \) is the angle in the \( xy \)-plane from the positive \( x \)-axis.
• \( \phi \) is the angle from the positive \( z \)-axis down to the point.

The transformation from spherical coordinates to Cartesian coordinates \((x, y, z)\) is given by:\[ x = \rho \cos \theta \sin \phi,\]\[ y = \rho \sin \theta \sin \phi,\]\[ z = \rho \cos \phi.\] These formulas help in transforming problems originally defined in rectangular coordinates into spherical coordinates, which can simplify the problem by leveraging the symmetry of the sphere.

Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of a composition of functions. When dealing with multivariable functions, the chain rule becomes a bit more complex but follows the same principle. It allows us to differentiate a composite function by taking derivatives of each function along the way. Here is a simple version for two variables:If you have a function \( w = f(x, y) \), and both \( x \) and \( y \) depend on another variable \( t \), then\[ \frac{dw}{dt} = \frac{\partial f}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dt}.\]In the context of our problem, we used the chain rule to differentiate \( w = x^2 y + z^2 \) concerning \( \theta \), considering the relationship between \( x, y, z \) and \( \theta \). This is essential for understanding how changes in \( \theta \) affect \( w \).

Product Rule

The product rule is another essential tool for finding derivatives, especially when dealing with products of functions. In essence, the product rule states that the derivative of a product of two functions is given by:\[ \frac{d}{dx}[u(x)\cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x).\]When applied in contexts involving multiple variables, it helps differentiate terms that are products of two or more functions of the same variable. For example, in our original exercise, to differentiate \((\cos^2 \theta)(\sin \theta)\), we:

• First differentiated \( \cos^2 \theta \), then multiplied by \( \sin \theta \).
• Then, differentiated \( \sin \theta \) and multiplied by \( \cos^2 \theta \).

This systematic approach using the product rule ensures every part of the product is properly accounted for in the derivative.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of several variables. Unlike single-variable calculus, multivariable calculus deals with functions that have inputs and outputs in more than one dimension.Some key concepts include:

• Partial derivatives, which measure how a function changes as one specific variable changes, holding the others constant.
• The chain rule for multivariable functions, which lets us take derivatives of compositions involving multiple variables.
• Tools such as gradient, divergence, and curl, which provide additional ways of describing and analyzing these functions.

In the specific problem we solved, multivariable calculus enabled us to consider \( w(x, y, z) \) where \( x, y, z \) are functions of \( \rho, \theta, \phi \). This situation requires understanding and applying each of these tools—especially the partial derivative—to compute how \( w \) changes concerning \( \theta \). This comprehensive approach is key to tackling problems that involve variables describing three-dimensional space.