9.2 The Directional Derivative 1)gradients 2)Gradient at a point 3)Vector differential operator.

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Presentation transcript:

9.2 The Directional Derivative 1)gradients 2)Gradient at a point 3)Vector differential operator

REV:Vectors in 2D & 3D Unit vector = length = Dot product =

Example1 gradients Comput for = del f = grad f Gradient

Example2 If find at (2,-1,4) Gradient at a point

Vector Differential Operator 2D 3D

Directional Derivative. Example3 Find the directional derivative of. at (1,1) in the direction of (A) the unit vector (B) a unit vector in the direction of 3i+4j ….(C) a unit vector whose angle with the positive x-axis is (D) The unit vector 0i+j

Directional Derivative. Geometric representation (1D)

Generalization of Partial Diff In this section we will introduce a type of derivative, called a directional derivative. Suppose that we wish to find the directional derivative of f at (x0,y0) in the direction of an arbitrary unit vector u = To do this we consider the surface S with equation z=f(x,y) And we let z0=f(x0,y0) then the point P(x0,y0,z0) lies on S. The vertical plane that passes though P in the direction of u intersects S in a curve C. The slope of the tangent line T to C at P is the directional derivative of f at (x0,y0) in the direction of u See this link See this

Directional Derivative. Example5 Find the directional derivative of. at (1,-1,2) in the direction of:

Directional Derivative. ezsurf('4*x^2+y^2',[0,4])

Example4