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, xn) = 0, where F is a polynomial. Finding the gradient of a curveTo find the gradient of a curve, you must draw an accurate sketch of the curve. For example, a level surface in three-dimensional space is defined by an equation of the form F(x, y, z) = c. Find the perfect one or modify them to fit your needs.
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Definition from W3C
Gradient is another word for “slope”. A value of 0deg is equivalent to
to top. 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). At a non-singular point, it is a nonzero normal vector.
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Using the convention that vectors in
R
n
{\displaystyle \mathbb {R} ^{n}}
are represented by column vectors, and that covectors (linear maps
n
R
{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} }
) are represented by row vectors,a the gradient
f
{\displaystyle \nabla f}
and the derivative
d
f
{\displaystyle df}
are expressed as a column and row vector, respectively, with the same components, but transpose of each other:
While these both have the same components, they differ in what kind of mathematical object they represent: at each point, the derivative is a cotangent vector, a linear form (covector) which expresses how much the (scalar) output changes for a given infinitesimal change in (vector) input, while at each point, the gradient is a tangent vector, which represents an infinitesimal change in (vector) input.
If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms look at more info the normalized bases, which we refer to as
e
{\displaystyle {\hat {\mathbf {e} }}_{i}}
and
e
i
{\displaystyle {\hat {\mathbf {e} }}^{i}}
, using the scale factors (also known as Lamé coefficients)
h
i
=
e
i
=
g
i
i
=
1
/
e
i
{\displaystyle h_{i}=\lVert \mathbf {e} _{i}\rVert ={\sqrt {g_{ii}}}=1\,/\lVert \mathbf {e} ^{i}\rVert }
:
where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. .