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Homework Statement
can there be a vector field G on R^3 such that G=<xsiny, cosy, z-xy>?
Homework Equations
The Attempt at a Solution
\the answer is no, but i don't understand why. any help is appreciated, thanks
No, a vector field G cannot exist on all points in R^3 because R^3 is an infinite space and there is no way to define a vector at every single point.
Yes, it is possible for a vector field G to be continuous on R^3. In fact, many real-world physical phenomena can be described using continuous vector fields.
In order for a vector field G to exist on R^3, it must satisfy the three-dimensional vector calculus equations, such as the divergence and curl equations. Additionally, it must have a well-defined direction and magnitude at each point in R^3.
To determine if a vector field G exists on a specific subset of R^3, we can use the divergence theorem or Stokes' theorem to evaluate the flux or circulation of the vector field over the boundary of the subset. If the value is non-zero, then a vector field G exists on that subset.
Yes, vector fields on R^3 have many real-world applications in fields such as physics, engineering, and fluid dynamics. They are used to model and understand the behavior of electric and magnetic fields, fluid flow, and other physical phenomena.