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Chapter 14: Problem 5

Interpret the curl of a general rotation vector field.

Short Answer

Expert verified
Answer: The curl of the general rotation vector field F(x, y, z) = (-y, x, 0) is given by curl(F) = (0, 0, x + y). This means that the rotation of the vector field is only in the z-direction, and the magnitude of the rotation increases with the distance from the origin.

Step by step solution

01

Define a general rotation vector field

A general rotation vector field can be represented as: F(x, y, z) = (-y, x, 0) This vector field has the property that the closer a point is to the origin, the slower it rotates.
02

Calculate the curl of the vector field

The curl of a vector field F is a vector that represents the rotational behavior of F at each point. It is defined by the following cross product with the differential operator: curl(F) = 鈭� 脳 F Where 鈭� is the differential operator (del), represented as: 鈭� = (鈭�/鈭倄, 鈭�/鈭倅, 鈭�/鈭倆) Given our vector field F(x, y, z) = (-y, x, 0), we can calculate the curl as follows: curl(F) = (鈭� 脳 F) = ( (鈭�/鈭倄, 鈭�/鈭倅, 鈭�/鈭倆) 脳 (-y, x, 0) ) Now, let's compute the cross product: curl(F) = ( (鈭�/鈭倅 * 0 - 鈭�/鈭倆 * x), (鈭�/鈭倆 * (-y) - 鈭�/鈭倄 * 0), (鈭�/鈭倄 * x - 鈭�/鈭倅 * (-y)) ) This simplifies to: curl(F) = (0, 0, x + y)
03

Interpret the curl of the general rotation vector field

The result of curl(F) = (0, 0, x + y) tells us that the rotation is only in the z-direction, which is the third component of the curl vector. The rotation increases with the distance from the origin, as x and y increase. In conclusion, the curl of a general rotation vector field F(x, y, z) = (-y, x, 0) is given by curl(F) = (0, 0, x + y). This means that the rotation of the vector field is only in the z-direction, and the magnitude of the rotation increases with the distance from the origin.

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Most popular questions from this chapter

Prove Green's Second Identity for scalar-valued functions \(u\) and \(v\) defined on a region \(D\) : $$\iiint_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d V=\iint_{S}(u \nabla v-v \nabla u) \cdot \mathbf{n} d S$$ (Hint: Reverse the roles of \(u\) and \(v\) in Green's First Identity.)

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