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How does vector calculus apply in fluid mechanics?
The velocity at each point in the fluid is a vector. If the fluid is compressible, the divergence of the velocity vector is nonzero in general. In a vortex the curl is nonzero.
Definition of Divergence of a vector field?
hedivergence of a vector fieldF= (F(x,y),G(x,y)) with continuous partial derivatives is defined by:
What is physical significance of divergence?
Divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point.
Examples of divergence of a vector field?
I am not sure if this is the answer you are looking for, since the question is listed in both Physics and Abstract Algebra, so I will try to give you some examples from physics. One of the indicators of a divergence of a vector field is the presence of a source. For example the electric field can be represented by a vector field, with each vector pointing along the field and has a length proportional to the strength of the electric field at that position. A point source then causes an electric field with a divergence at the location of the point source, with the vectors all pointing away from it (positive charge) or towards it (negative charge). Another example would be some point mass and the Newtonian gravitational field. One of Maxwell's equations states that the magnetic field cannot have any divergences meaning that there are no magnetic monopoles.
What is curl in mathematical terms?
Curl represents the force of rotation in a 3-D vector field. Generally, the curl vector at a given point is the answer to the question, "What would happen if I stuck something there that could spin but couldn't move?" Unless the curl is zero, it would spin perpendicularly to the curl vector (according to the right-hand rule), and the longer the vector is, the faster. Curl is mathematically defined in a given direction as the limit of "circulation over area", i.e. the line integral of a circle around the point, divided by the area of the circle, with the circle shrinking towards the point. More practically, the actual vector can found by taking the cross product of the gradient operator with the function that defines the field: curl_x = ∂F/∂y - ∂F/∂z curl_y = ∂F/∂z - ∂F/∂x curl_z = ∂F/∂x - ∂F/∂y
