Directional Derivative of w on Intersection of Surfaces

  • Thread starter alvielwj
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In summary, the directional derivative of w = x^3 + y^2 + 3z at the point (1, 3, 2) on the curve of intersection of the surfaces is found using the cross product of the two gradient vectors.
  • #1
alvielwj
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Homework Statement


Find the directional derivative of w = x^3 + y^2 + 3z at the point (1, 3, 2) on the curve of intersection of the surfaces
x^2 + y^2− 2xz = 6, 3x^2− y^2 + 3z = 0, in the direction of increasing z.


Homework Equations





The Attempt at a Solution


i know how to solve the directional dirivative of a function of at a point on the given vector.but in this question I don't know how to solve the equations,and "in the direction of increasing z" does not make sense
 
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  • #2
The tangent vector along the curve is perpendicular to the gradient vectors of the two surfaces. Find them at (1,3,2). Now that you have these two vectors, how do you find one that is perpendicular to them both?
 
  • #3
cross product of these two vectors
 
  • #4
But I can't get the meaning of "in the direction of increasing z"
 
  • #5
alvielwj said:
But I can't get the meaning of "in the direction of increasing z"

Using your cross product you'll get a vector t for the tangent vector. But -t is also a tangent. Pick the one that has positive z component.
 
  • #6
I want to get the gradient vector of x^2 + y^2− 2xz = 6,so z=(x^2+y^2-6)/2x
,is the gradient vector (dz/dx dz/dy) but i should have three dimensions?
 
  • #7
If f=x^2+y^2-2xz-6=0 is your surface the gradient vector is (df/dx,df/dy,df/dz) (all partial derivatives). Of course you get three components.
 
  • #8
Why do we calculate the gradient vectors of the surfaces? What is the geometrical meaning of gradient vector? I think I should calculate the normals of two surfaces and get the cross product of two normals, because the vector of the intersection curve is perpendicular to both of the normals..
 
  • #9
The normal vector points in the same direction as the gradient vector. Don't you calculate the normal vector using the gradient vector??
 
  • #10
AH, finally,i figured it out , thank a lot
 

What is a directional derivative?

A directional derivative is a type of derivative that measures the rate of change of a function with respect to a specific direction in a multi-dimensional space. It allows us to determine how a function changes along a specific direction from a given point.

How is a directional derivative calculated?

The directional derivative is calculated using the dot product of the gradient of the function and the unit vector in the direction of interest. The resulting value is the slope of the tangent line to the function in the given direction.

What is the relationship between directional derivatives and partial derivatives?

The directional derivative is a generalization of the partial derivative. While the partial derivative measures the rate of change of a function in a specific direction, the directional derivative measures the rate of change in any direction. The partial derivatives of a function at a point can be used to calculate the directional derivative in any direction at that point.

How is a directional derivative used in real-world applications?

Directional derivatives are used in many areas of science and engineering, such as physics, economics, and computer graphics. They can be used to optimize functions in various directions, determine the direction of maximum change, and analyze the sensitivity of a system to changes in different directions.

What is the physical interpretation of a directional derivative?

The directional derivative can be interpreted as the rate of change of a function in a specific direction, similar to how a regular derivative measures the rate of change of a function at a specific point. It can also be thought of as the slope of a tangent line to the function in the given direction.

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