I have a placement exam in university tomorrow that will allow me to skip my entire calculus requirement. The only part of the included syllabus that I have no covered are "vector fields".
What are these, (I know that they have functions..?).
Where can I learn the theory behind them, and where they fit into math, (or how to solve problems involving them with good explanations).|||a vector field is (loosely speaking) a function from R^n to R^n (usually n = 2 or 3).
vector fields are often defined by their coordinate functions, for example, in 3 dimensions, you would write:
F(x,y,z) = (F1(x,y,z),F2(x,y,z),F3(x,y,z))
where each of the F1,F2 and F3 are real-valued functions of 3 variables.
the functions F1,F2, and F3 are in turn, examples of what are sometimes called "scalar fields".
a scalar field is just a function F:R^n --%26gt; R ( a real-valued function of n variables).
since a real-valued function of n variables has n possible partial derivatives, we can form a natural vector field from any scalar field called the gradient:
grad F = (∂F/∂x1, ∂F/∂x2,...,∂F/∂xn)
(the gradient is often indicated with an upside-down delta sign).
the gradient is a generalization of the derivative for functions of more than one variable: at every point of a scalar field, the gradient points in the direction of greatest rate of change of the scalar field, and the magnitude of the gradient is the greatest rate of change.
just as one might ask whether there exists a function whose derivative is f(x), for a given function f:R--%26gt;R (which is what we do when we integrate), one can ask if, given a vector field F:R^n--%26gt;R^n, whether or not that vector field is the gradient of some scalar field.
honestly though, if you haven't been exposed to this material, i'm not entirely sure if it is a good idea for you to skip your calculus courses, unless you never plan to use math again.
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