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Chapter 18: Problem 3

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

Expert verified
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

01

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).
02

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.

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Most popular questions from this chapter

Berechnen Sie von den Vektorfeldern \(\vec{F}\) die Divergenz. Geben Sie an, wo Quellen und Senken liegen, bzw. wo das Feld quellen- und senkenfrei ist. a) \(\vec{F}(x, y, z)=(x-a, y, z)\) b) \(\vec{F}(x, y, z)=\left(a,-x, z^{2}\right)\)

Berechnen Sie für das Vektorfeld \(\vec{F}(x, y, z)=(0, y, z)\) das Linienintegral längs des Weges \(C\) vom Punkt \((0,0,0)\) zum Punkt \((0,0,3)\),

Sind die Vektorfelder wirbelfrei? a) \(\vec{F}(x, y, z)=(a, x, b)\) b) \(\vec{F}(x, y, z)=\frac{(x, y, z)}{x^{2}+y^{2}+z^{2}}\)
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