Define Scalar and Vector Field.
ANSWER : A scalar field is a concept in mathematics that assigns a single scalar value to every point in space. Scalars are quantities that have only magnitude (size) and no direction. For example, temperature can be represented as a scalar field, where each point in space has a specific temperature value.
A vector field, on the other hand, assigns a vector quantity to every point in space. Vectors have both magnitude and direction. A common example of a vector field is the velocity field of a fluid flow, where each point in space has a specific velocity vector associated with it.
In mathematical terms, a scalar field is denoted by the function f(x, y, z), where (x, y, z) represents the coordinates of a point in three-dimensional space. The value of the scalar field at a specific point (x, y, z) is given by f(x, y, z).
A vector field is represented by a vector function F(x, y, z) = ⟨f₁(x, y, z), f₂(x, y, z), f₃(x, y, z)⟩, where f₁, f₂, and f₃ are scalar functions that determine the components of the vector at each point (x, y, z). The vector field F(x, y, z) can be visualized as a set of vectors attached to each point in space, indicating both magnitude and direction.
In summary, a scalar field assigns a scalar value to each point in space, while a vector field assigns a vector value to each point in space.
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