Question

Berechnen Sie das Linienintegral \(\oint \vec{F} \cdot \overrightarrow{d s}\) längs des Rechtecks in der \(y\) - \(z\)-Ebene mit den Seiten \(a\) und \(b\). ??- ben durch \(\vec{F}(x, y, z)=5(0, y, z)\)

Short answer

The line integral of the vector field \( \vec{F}(x, y, z)=5(0, y, z) \) along the rectangle in the y-z plane with sides a and b is zero.

Step-by-step solution

Defining the Path

The line integral is calculated over the perimeters of the rectangle in the y-z plane with sides of lengths a and b. This rectangle can be defined by 4 line segments: from (0,0) to (0,b), from (0,b) to (a,b), from (a,b) to (a,0) and from (a,0) to (0,0).

Calculating the Line Integral

The process of calculating the line integral involves determining the dot product of the vector field and the differential arc length, ds, over each of the 4 segments and then adding these up. However, the given vector field has only y and z-components. As we traverse each segment in the x-direction, y and z are constants and thus their differentials, dy and dz, are zero. Therefore, along each segment, the dot product of the vector field and ds is zero, and as a result, the line integral over the rectangle is zero.

Key concepts

Understanding Vector Calculus

Vector Calculus is a branch of mathematics concerned with integrating and differentiating vector fields. It extends traditional calculus into multi-dimensional spaces.
When we deal with vector calculus, we are often confronted with operations that involve vectors, like gradients, divergences, and curls. These operations are applied to functions that map \((x, y, z)\) coordinates into vectors.

• The concept helps in fields like physics, where it describes electromagnetic forces.
• It's used to represent and analyze changes along a dimension or within a field.

For line integrals, we particularly focus on the way these vectors interact with a path in space. In our case, the vector field specifies force vectors influencing a particle moving around a rectangle in the y-z plane.

Exploring Vector Fields

Vector Fields are mathematical constructs where each point in a portion of space is associated with a vector. These are crucial for understanding environments where multiple directions and magnitudes are in play.
The vector field given, \( \vec{F}(x, y, z) = 5(0, y, z) \), indicates that no force exists along the x-direction as the x-component is zero.

• Each point in the plane carries directional vector \((0, 5y, 5z)\), highlighting how y and z-components play roles.
• Conceptually, it describes a scenario possibly like fluid flow where movement is constrained to the y-z plane.

Understanding vector fields allows us to draw insights into how particles might move throughout space, influenced by these vectors.

Dot Product in Line Integrals

The Dot Product is an algebraic operation used to find the cosine of the angle between two vectors, thus calculating how much one vector goes in the direction of another. In line integrals, it is essential for relating the vector field to the path.
In our case, the dot product \( \vec{F} \cdot \overrightarrow{d{s}} \) is essential for calculating the interaction of our field with a path represented as little segments of the rectangle.

• If the vectors are perpendicular, like in segments along the x-axis exclusively, the dot product becomes zero.
• This is why, in this exercise, the line integral along the y-z plane rectangle results in zero: the displacement vectors are not contributing to any work against \( \vec{F} \).

Path Integration Made Simple

Path Integration, or line integration, is a key operation where we sum up the products of a vector field with an element of distance, along a specific path.
It examines how a field exerts influence across a curved or straight path, such as the perimeter of a rectangle. For our rectangle, since the vector field \(\vec{F}\) has no x-component:

• The path segments along x show negligible contribution, as dy and dz are zero.
• This simply means traversing those segments doesn't encounter any field force, nullifying the integral's value.

Therefore, sometimes a path integral simplifies amazingly due to properties of symmetry and vector orientation, yielding a zero value as in this exercise.