Curl of a Vector Field using Determinants

Consider the vector field F = ⟨(x²)y, -x/y, xyz⟩. Find the curl by taking the determinant of the matrix with i, j, k; d/dx, d/dy, d/dz; (x²)y, -x/y, xyz.

When approaching the problem of finding the curl of a vector field, we engage with several interconnected concepts in vector calculus. The curl of a vector field is an operation that describes the rotation of the field at a given point. In physical terms, if you Picture a small paddle wheel placed in the vector field, a non-zero curl would imply that the wheel would rotate. This is particularly important in fluid dynamics and electromagnetism where rotational forces play a critical role.

In this problem, we determine the curl using the determinant of a matrix constructed from unit vectors and partial derivatives. The first row of the matrix contains the unit vectors i, j, and k. The second row comprises the partial derivatives with respect to x, y, and z, respectively, and the third row consists of the components of the vector field. This configuration allows us to leverage the determinant to systematically compute the cross product of the differential operators with the vector field.

While solving, keep in mind the properties of determinants and partial derivatives. The determinant will be expanded along the top row, and requires careful attention to the sign changes that accompany cofactor expansion. Once the partial derivatives are taken and the cross products are computed, it is critical to check for simplifications that can reduce the complexity of the resulting vector field. This procedural approach underscores the synthesis of algebraic and geometric reasoning inherent in vector calculus problems.

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