Problem 8 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=x \cos y-y e^{x} $$

Step-by-Step Solution

Answer

a) \[ D_{11} f(x, y) = -y \, e^x \], b) \[ D_{22} f(x, y) = -x \, \cos y \], c) \[ D_{12} f(x, y) = D_{21} f(x, y) = -\sin y - e^x \].

Step 1 - Compute the first partial derivative with respect to x

Calculate the partial derivative of the function with respect to x. Given the function \[ f(x, y) = x \, \cos y - y \, e^x \], we find \[ f_x(x, y) = \frac{\partial}{\partial x} \, (x \, \cos y - y \, e^x) = \cos y - y \, e^x. \]

Step 2 - Compute the second partial derivative with respect to x

Take the partial derivative of the function we just obtained in Step 1 with respect to x again:\[ f_{xx}(x, y) = \frac{\partial}{\partial x} \, (\cos y - y \, e^x) = -y \, e^x. \]

Step 3 - Compute the first partial derivative with respect to y

Calculate the partial derivative of the function with respect to y. Given the function \[ f(x, y) = x \, \cos y - y \, e^x \], we find \[ f_y(x, y) = \frac{\partial}{\partial y} \, (x \, \cos y - y \, e^x) = -x \, \sin y - e^x. \]

Step 4 - Compute the second partial derivative with respect to y

Take the partial derivative of the function we just obtained in Step 3 with respect to y again:\[ f_{yy}(x, y) = \frac{\partial}{\partial y} \, (-x \, \sin y - e^x) = -x \, \cos y. \]

Step 5 - Compute the mixed partial derivative D12

Find the partial derivative of \(f_x(x,y)\) with respect to y:\[ f_{xy}(x, y) = \frac{\partial}{\partial y} \, (\cos y - y \, e^x) = -\sin y - e^x. \]

Step 6 - Compute the mixed partial derivative D21

Find the partial derivative of \(f_y(x,y)\) with respect to x:\[ f_{yx}(x, y) = \frac{\partial}{\partial x} \, (-x \, \sin y - e^x) = -\sin y - e^x. \]

Step 7 - Confirm equality of mixed partial derivatives

We have shown that \[ f_{xy}(x, y) = -\sin y - e^x \] and \[ f_{yx}(x, y) = -\sin y - e^x, \] thus confirming \[ f_{xy}(x, y) = f_{yx}(x, y). \]

Key Concepts

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions of multiple variables. This includes functions of two or more variables like our given function \[ f(x, y) = x \, \cos y - y \, e^x \]. In multivariable calculus, we often consider partial derivatives. These are derivatives of functions with respect to one variable while holding the others constant. This is crucial because it allows us to study changes in the function in one direction at a time.

• For example, the partial derivative \( f_x \) represents how \( f \) changes as \( x \) changes while keeping \( y \) constant.
• Similarly, \( f_y \) represents how \( f \) changes as \( y \) changes while keeping \( x \) constant.

Understanding these concepts helps in solving problems involving rates of change in multiple dimensions.

Mixed Partial Derivatives

Mixed partial derivatives involve taking the partial derivatives of a function with respect to two different variables. For example, we have\[ f_{xy}(x, y) = \frac{\partial^2 f}{\partial y \partial x} \] and \[ f_{yx}(x, y) = \frac{\partial^2 f}{\partial x \partial y}.\]Fascinatingly, under certain conditions, these mixed partial derivatives are equal. These conditions are usually met if the function and its first partial derivatives are continuous. This is called Schwarz's theorem or the symmetry of second derivatives, and it is very useful. It simplifies calculations and is used to verify the consistency of solutions in multivariable calculus problems.In our specific problem, we've seen that\[ f_{xy}(x, y) = f_{yx}(x, y).\]This symmetry can provide valuable insights into the nature of the function and its behavior at different points.

Analytic Geometry

Analytic geometry facilitates understanding geometric concepts using algebra and calculus. In the context of multivariable calculus, it allows us to visualize functions of two or more variables in a geometric manner, often as surfaces or curves in three-dimensional space. Consider our function again. It represents a surface in 3D space:\[ f(x, y) = x \cos y - y e^x.\]Every point \((x, y)\) on this surface has a corresponding value of \( f(x, y) \), which shows us how the surface bends and twists.

• The partial derivatives \( f_x \) and \( f_y \) give us the slopes of the surface in the \( x \) and \( y \) directions, respectively.
• Second partial derivatives can tell us about the curvature and the concavity of the surface.

In summary, using analytic geometry in multivariable calculus enables us to link algebraic functions with their geometric representations, offering deeper insights and more powerful methods for analysis.