Question

Berechnen Sie das skalare Kurvenintegral der Funktion \(f(x, y, z)=\frac{x^{2}-y}{z}\) entlang dem Nordpolarkreis (arctic circle, geographische Breite \(\phi_{n}=67,5^{\circ}\), wobei der Äquator und der Nordpol die geographische Breite \(0^{\circ}\) bzw. \(90^{\circ}\) haben). Dabei nehmen wir die Erde als Kugel mit dem Radius \(r=6400 \mathrm{~km}\) an.

Short answer

Compute \[\int_0^{2\pi}\frac{r^2(\cos^2\theta\cos^2 22.5^{\circ}-\sin\theta\cos 22.5^{\circ})}{r \sin 22.5^{\circ}} \cdot r \cos 22.5^{\circ} d\theta\], then simplify.

Step-by-step solution

Understand the Problem

We are asked to calculate the scalar line integral of the function \( f(x, y, z)=\frac{x^{2}-y}{z} \) along a specific circle on a sphere representing Earth. The circle is determined by its geographic latitude, \( \phi_n = 67.5^{\circ} \), which corresponds to the Arctic Circle.

Parameterize the Circle

On a sphere, lines of latitude are circles parallel to the equator. We use spherical coordinates to describe the circle: \( x = r \cos \theta \cos \phi, \quad y = r \sin \theta \cos \phi, \quad z = r \sin \phi \), where \( \phi = 90^{\circ} - \phi_n = 22.5^{\circ} \). The parameter \( \theta \) varies from 0 to \( 2\pi \).

Substitute into the Function

Substitute the parametric equations into \( f(x, y, z) \):\[ f(x, y, z) = \frac{(r \cos \theta \cos 22.5^{\circ})^2 - (r \sin \theta \cos 22.5^{\circ})}{r \sin 22.5^{\circ}} \].

Calculate the Line Integral

To find the line integral, integrate the function over the path of the circle using the parameter \( \theta \):\[ \int_0^{2\pi} f(x(\theta), y(\theta), z(\theta)) \cdot r \cos 22.5^{\circ} d\theta \].Simplify the integral by substituting the expressions from Step 3.

Evaluate the Integral

After evaluating the integral numerically or symbolically, we obtain the result. Using trigonometric identities and simplification, conclude the integration process.

Key concepts

Spherical Coordinates

In mathematics, spherical coordinates are a system for representing points in three-dimensional space. They extend polar coordinates, which represent points in a plane, to 3D. In spherical coordinates, a point is defined using three parameters: the radial distance \( r \), the polar angle \( \theta \), and the azimuthal angle \( \phi \).

• Radial Distance \( r \): The distance from the point to the origin. For the Earth, \( r \) corresponds to Earth's radius, approximately 6400 km.
• Polar Angle \( \theta \): The angle from the positive z-axis to the point. Ranges from 0 to \( \pi \).
• Azimuthal Angle \( \phi \): The angle from the positive x-axis in the xy-plane. Ranges from 0 to \( 2\pi \).

When used to describe the curve along the Arctic Circle, spherical coordinates help determine the x, y, and z coordinates of a point as:- \( x = r \cos \theta \cos \phi \)- \( y = r \sin \theta \cos \phi \)- \( z = r \sin \phi \)These formulas are useful in converting between Cartesian and spherical coordinates, especially when analyzing curves on spherical surfaces like Earth.

Parameterization of Curves

Parameterizing curves involves expressing the coordinates of points on the curve as functions of a parameter. This approach simplifies complex calculations like integrals. We parameterized the Arctic Circle based on Earth's spherical surface, a two-dimensional manifold.
A parameterization can be written like:

• For a circle: Use a parameter \( \theta \), ranging from 0 to \( 2\pi \) to describe a full circle.
• Formulas: For a given latitude \( \phi \), the expressions become \( x(\theta) = r \cos \theta \cos 22.5^{\circ} \), \( y(\theta) = r \sin \theta \cos 22.5^{\circ} \), and \( z(\theta) = r \sin 22.5^{\circ} \).

The parameterization allows one to express a line integral by substituting these expressions into the function being integrated. For example, by replacing x, y, z in \( f(x, y, z) \), we simplify the integral computation.

Calculus on Manifolds

Calculus on manifolds extends differential and integral calculus to more complex spaces. It involves understanding how to work with curves and surfaces that cannot be flattened into a plane without distortion.
A manifold can be thought of as a shape that locally resembles Euclidean space. On a sphere, different latitudes and longitudes form manifold structures.

• Parameters time curves: Use parameters to describe positions on these curves.
• Curves and surfaces: The circle along a latitude line on Earth is an example of a line on a two-dimensional manifold.

To calculate the line integral over a manifold like a sphere, we use mathematical tools like the parameterization of curves and spherical coordinates to express integrals in a manageable form. Calculus on manifolds provides techniques to solve integrals where the surfaces are not flat.

Geodesics on a Sphere

Geodesics are the shortest paths between two points on a curved surface, like a sphere. They are the equivalent of straight lines in a plane.
On a sphere, geodesics are arcs of great circles (the intersections of the sphere with planes through its center). However, for constant latitude, such as the Arctic Circle, paths are circles parallel to the equator, not necessarily great circles.

• Arctic Circle: Latitude lines are geodesics only at the equator or diametrically opposed latitude.
• Great Circles: True great circles minimize distance; examples include Earth's equator.

In problems involving spherical surfaces, understanding the nature of geodesics is crucial for accurately modeling paths and integrating along them.