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2.6.7. Directional derivative, gradient

Lecture



Gradient function   2.6.7.  Directional derivative, gradient at the point M (x, y) is called a vector with the origin at the point M , with its coordinates the partial derivatives of the function z ,   2.6.7.  Directional derivative, gradient . The symbol for the gradient is often used.   2.6.7.  Directional derivative, gradient . The gradient indicates the direction of the fastest growth function at a given point.

Derivative function   2.6.7.  Directional derivative, gradient at point M (x, y) in the direction of the vector   2.6.7.  Directional derivative, gradient called   2.6.7.  Directional derivative, gradient .

If the function   2.6.7.  Directional derivative, gradient differentiable, then the derivative in this direction is calculated by the formula   2.6.7.  Directional derivative, gradient , where α is the angle between the vector s and the axis OX .

Using the definition of the gradient, the formula for the derivative in the direction can be given the form:   2.6.7.  Directional derivative, gradient , where the vector s o is the ort of the vector s , i.e. the derivative of the function in this direction is equal to the scalar product of the gradient of the function and the unit vector of this direction.

Derivative   2.6.7.  Directional derivative, gradient in the direction of the gradient has the greatest value of   2.6.7.  Directional derivative, gradient .


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Mathematical analysis. Differential calculus

Terms: Mathematical analysis. Differential calculus