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Definition/Summary
The tangent to a curve in a plane at a particular point has the same Gradient as the curve has at that point.
More generally, the (n-1)-dimensional tangent hyperplane to an (n-1)-dimensional surface in n-dimensional space at a particular point has the same Gradient as the surface has at that point.
So if [itex]A\,=\,(a_1,a_2,\cdots a_n)[/itex] is a point on a surface defined by the equation [itex]F(x_1,x_2,\cdots x_n) = 0[/itex], then the tangent hyperplane to the curve through [itex]A[/itex] is [itex]\frac{\partial F}{\partial x_1}\arrowvert_A(x_1 – a_1)\,+\,\frac{\partial F}{\partial x_2}\arrowvert_A(x_2 – a_2)\,+\,\cdots\,\frac{\partial F}{\partial x_2}\arrowvert_A(x_n – a_n)\,=\,0[/itex]
If a curve in n dimensions is defined using a parameter t as [itex]A(t)\,=\,(a_1(t),a_2(t),\cdots a_n(t))[/itex] , then its tangent is:
[itex](x_1 – a_1) / \frac{da_1}{dt}\,=\,(x_2 – a_2) / \frac{da_2}{dt}\,=\,\cdots\,=\,(x_n – a_n) / \frac{da_n}{dt}[/itex]
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