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In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane (for instance), can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane.
The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals).
Gradient of a vector field is intuitively the Flux/volume leaving out of the differential volume dV. For a scalar field(say F(x,y,z) ) it represents the rate of change of F along the the 3 perpendicular ( also called orthonormal ) vectors you defined your system with (say x, y, z ).
The gradient is a fancy word for derivative, or the rate of change of a function. It's a vector (a direction to move) that. Points in the direction of greatest increase of a function (intuition on why) Is zero at a local maximum or local minimum (because there is no single direction of increase)
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