What if ∂Q/∂x = ∂P/∂y, but ∂Q/∂z ≠ ∂R/∂y ?Bill Nye Tho said:Homework Statement
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Homework Equations
∇ x F = 0
∂Q/∂x = ∂P/∂y
The Attempt at a Solution
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Given that no sketch of the vector field is given;
Is determining the curl of a vector field the most fail proof of determining whether it is conservative?
I'm just wondering whether or not determining ∂Q/∂x = ∂P/∂y is just as fail proof (Given that: F=Pi + Qj + Rk) because it seems like a faster method within the boundary of this course.
SammyS said:What if ∂Q/∂x = ∂P/∂y, but ∂Q/∂z ≠ ∂R/∂y ?
I've not aware of that sort of distinction.Bill Nye Tho said:Also, the answer to your question would be that the field would only be conservative in the XY plane but not in the XZ or YZ.
Vector analysis is a branch of mathematics that deals with the study of quantities that have both magnitude and direction, known as vectors. It is important because it allows us to analyze and understand complex physical phenomena, such as fluid flow and electromagnetic fields.
A vector field is conservative if the work done by the field on a particle moving along a closed path is zero. Mathematically, this means that the curl of the vector field must be equal to zero. If the curl is not zero, the vector field is non-conservative.
A conservative vector field has the property that the work done by the field on a particle depends only on the endpoints of the path and not on the path itself. This makes it easier to calculate work and other physical quantities, as well as simplifying mathematical calculations.
A conservative vector field can be expressed using a scalar potential function. This function can be found by taking the line integral of the vector field along any path from a fixed point to a variable point. The resulting function is called the potential function and is unique up to a constant.
Vector analysis has many real-world applications in various fields such as engineering, physics, and economics. Some examples include analyzing fluid flow in pipes, calculating the work done by magnetic fields on charged particles, and modeling stock market trends.