Question

Berechnen Sie die partiellen Ableitungen \(f_{x x}, f_{x y}, f_{y x}\) und \(f_{y y}\) der Funktion $$ z=R^{2}-x^{2}-y^{2} $$

Short answer

The second order partial derivatives are \( f_{xx} = -2 \), \( f_{xy} = 0 \), \( f_{yx} = 0 \) and \( f_{yy} = -2 \)

Step-by-step solution

Compute \( f_{x} \)

Differentiate the given function once w.r.t variable \(x\). All the terms that are not involving \(x\) are treated as constants, so \(R^2\) gives 0 and \(-y^2\) gives 0. Differentiation of \(-x^2\) w.r.t \(x\) is \(-2x\). So, \( f_{x} = \frac{d}{dx} (R^2 - x^2 - y^2) = -2x \)

Compute \( f_{xx} \)

Differentiate \( f_{x} \) once again w.r.t \(x\) to get \( f_{xx} \). So, \( f_{xx} = \frac{d}{dx} (-2x) = -2 \)

Compute \( f_{y} \)

Differentiate the given function once w.r.t variable \(y\). All the terms not involving \(y\) are treated as constants, so \(R^2\) gives 0 and \(-x^2\) gives 0. Differentiation of \(-y^2\) w.r.t \(y\) is \(-2y\). So, \( f_{y} = \frac{d}{dy} (R^2 - x^2 - y^2) = -2y \)

Compute \( f_{xy} = f_{yx} \)

Differentiate \( f_{x} \) w.r.t \(y\) to get \( f_{xy} \) and differentiate \( f_{y} \) w.r.t \(x\) to get \( f_{yx} \). As there is no \(y\) term in \( f_{x} \) and no \(x\) term in \( f_{y} \), both these second order partial derivatives are equal to 0. So, \( f_{xy} = f_{yx} = 0 \)

Compute \( f_{yy} \)

Differentiate \( f_{y} \) once again w.r.t \(y\) to get \( f_{yy} \). So, \( f_{yy} = \frac{d}{dy} (-2y) = -2 \)

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves calculating the rate at which a function changes. When we differentiate a function, we find its derivative, which gives us information about the function's slope at any given point.

In this exercise, we are asked to compute the partial derivatives of a function with respect to its variables \(x\) and \(y\). This means that we hold the other variables constant and only focus on the variable of interest.

• For example, to find \( f_{x} \), we treat variables other than \(x\) as constants.
• Similarly, for \( f_{y} \), we treat \(x\) as a constant and differentiate with respect to \(y\).

Partial differentiation is a crucial tool because it allows us to analyze how changes in one variable affect the function while keeping others constant. This is especially useful in multivariable functions, where multiple inputs contribute to an output.

Multivariable Calculus

Multivariable calculus extends the operations of calculus to functions of several variables. In our exercise, the function \( z = R^2 - x^2 - y^2 \) is dependent on two variables, \(x\) and \(y\).

When dealing with multivariable calculus, partial derivatives help us understand how each variable influences the function. We can explore surface properties and how gradients change within space.

• The second-order derivatives like \( f_{xx} \), \( f_{yy} \), and \( f_{xy} \) describe the curvature or concavity of the surface.
• For example, \( f_{xx} \,=\, -2 \) indicates a constant concavity in the direction of \(x\).
• The mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) show the rate at which one partial derivative changes with another variable. In our case, they are zero, reflecting the function's symmetrical nature about its axes.

Understanding these concepts not only helps solve mathematical problems but also offers insights into physical phenomena.

Mathematical Physics

In mathematical physics, the application of calculus, particularly multivariable calculus, helps in modeling and solving real-world problems. The function examined here resembles an expression for a paraboloid, a shape commonly found in physics. For instance, it can represent potential fields or surfaces of revolution.

Partial derivatives have significant implications, including:

• Describing how physical systems react to changes in different directions, such as temperature or pressure gradients.
• In fluid dynamics, partial derivatives help express velocity fields and dynamic changes in fluid flow.
• In electromagnetism, these derivatives are employed to analyze electric and magnetic fields, which are vector fields inherently involving multiple variables.

Through such practical utilizations, partial derivatives demonstrate their value in translating mathematical models into comprehendable physical phenomena. This bridging of theory and reality allows scientists and engineers to innovate and predict outcomes in complex systems.