The gradient is a vector-valued function, as opposed to a derivative, which is scalar-valued.
In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system.
- muti-variable function
-
scalar-valued function: Rn -> R, gradient vector, row matrix or column matrix.
f(x1, x2 ... xn): -
vector-valued functions: Rn -> Rm, mxn matrix. Jacobian matrix.
f1(x1, x2 ... xn)
f2(x1, x2 ... xn)
...
fm(x1, x2 ... xn)\
-
https://mathinsight.org/derivative_matrix
https://en.wikipedia.org/wiki/Gradient
- Linear approximation to a function
The gradient of a function f from the Euclidean space Rn to R at any particular point x0 in
Rn characterizes the best linear approximation to f at x0. The approximation is as follows:
f ( x ) ≈ f ( x0 ) + ( ∇ f ) x0 ⋅ ( x − x0 )
for x close to x0, where (∇f )x0 is the gradient of f computed at x0,
and the dot denotes the dot product on Rn.
This equation is equivalent to the first two terms in the multivariable Taylor series expansion of f at x0.
The approximation is valid only when the function is differentiable, and can be used with the points which is very close to x0.
it describes the tangent line or plane over that point.
This could be extended to multivariable linear approximation case.\