Directional derivatives, SIMPLE

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In summary, a directional derivative is a measure of the rate of change of a function in a specific direction at a given point. It is calculated by taking the dot product of the function's gradient and a unit vector in the desired direction. Directional derivatives have practical applications in fields such as physics, engineering, and economics. The SIMPLE method is a computational method that uses directional derivatives to solve fluid dynamics problems. The directional derivative can be negative, indicating a decrease in the function, positive, indicating an increase, or zero, indicating a constant function in the specified direction.
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pleasehelpme6
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f(x, y, z) = xe^y + ye^z + ze^x, at (0, 0, 0),

directional vector v = <-2, 0, 5>

i solved for gradient f = (e^y + ze^x, xe^y + e^z, ye^z + e^x), at f(0,0,0) to be...

gradient f = (1,1,1)

this would make the answer just be -2 + 0 + 5 = 3
but this isn't right.

can someone show me what i did wrong?
 
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please.
 
  • #3
The directional derivative involves the unit vector in the direction of v, not v itself. Find the unit vector u = v/|v| and use that in your calculation.
 
  • #4
thank you! that makes a lot of sense
 

FAQ: Directional derivatives, SIMPLE

What is a directional derivative?

A directional derivative is a measure of how a function changes in a specific direction. It provides information about the rate of change of a function in a particular direction at a specific point.

How is a directional derivative calculated?

The directional derivative is calculated by taking the dot product of the gradient of the function and a unit vector in the desired direction. This can also be written as the product of the gradient and the cosine of the angle between the gradient and the direction vector.

What is the significance of directional derivatives in real life?

Directional derivatives have many practical applications, such as in physics, engineering, and economics. For example, they can be used to determine the rate of change of temperature in a specific direction, or the rate of change of profit in a particular direction.

What is the SIMPLE method for calculating directional derivatives?

SIMPLE stands for "Sequentially-Implemented Multi-Purpose Program for Numerical Simulation of Fluid Flows". It is a computational method used to solve fluid dynamics problems, including calculating directional derivatives.

Can the directional derivative be negative?

Yes, the directional derivative can be negative. This indicates that the function is decreasing in the specified direction. A positive directional derivative indicates that the function is increasing in the specified direction, while a zero directional derivative indicates that the function is constant in that direction.

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