What is the Correct Directional Derivative for Vector w in the Given Scenario?

In summary, the conversation discusses how to compute the partial derivative using a given vector. The speaker mentions that they get a value of minus square root of five if they use the last vector, but are unsure if they should divide by the norm. Another person suggests that a direction derivative should be taken with respect to a unit vector, with a value of negative square root of five being correct. However, the speaker mentions that some authors do not require this and suggests using ||v|| instead of |v| for the magnitude of a vector. Ultimately, the correct value to consider depends on the definition of directional derivative provided in the book.
  • #1
DottZakapa
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i compute the partial derivative, the vector that i have to use the one in the text or
w=(2/(5^(1/2)), 1/(5^(1/2)))

using the last one i get minus square root of five , if i don't divide by the norm the answer should be B.
i don't understand what D means
 
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  • #2
DottZakapa said:
View attachment 255446
i compute the partial derivative, the vector that i have to use the one in the text or
w=(2/(5^(1/2)), 1/(5^(1/2)))

using the last one i get minus square root of five , if i don't divide by the norm the answer should be B.
i don't understand what D means

A direction derivative should be taken wrt a unit vector, so ##-\sqrt{5}## looks right. Unless ##\frac{\partial f}{\partial \vec v}## means something else?

The expression in ##D## is the magnitude of the gradient of ##f## at that point. Sometimes ##||u||## is used instead of ##|u|## for the "norm" or magnitiude of vector.
 
  • #3
PeroK said:
A direction derivative should be taken wrt a unit vector, so ##-\sqrt{5}## looks right. Unless ##\frac{\partial f}{\partial \vec v}## means something else?

The expression in ##D## is the magnitude of the gradient of ##f## at that point. Sometimes ##||u||## is used instead of ##|u|## for the "norm" or magnitiude of vector.

There are authors who do not ask for this. For example Munkres in "Analysis on manifolds" who defines the directional derivative for any non-zero vector.

Also, I believe it is better to use ##\Vert v \Vert## (Use \Vert v \Vert as tex command) instead of ##|| v ||## (formats a bit more nicely imo).
 
  • #4
so according to the text what shall i consider ?
 
  • #5
DottZakapa said:
so according to the text what shall i consider ?

Depends on your definition of directional derivative. So please give us the definition your book provides.

If it is w.r.t. a unit vector it is ##-\sqrt{5}##, otherwise ##-5## is correct.
 
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Related to What is the Correct Directional Derivative for Vector w in the Given Scenario?

What is a directional derivative?

A directional derivative is a measure of the instantaneous rate of change of a function in a specific direction. It is used to determine how much a function changes at a specific point in the direction of a given vector.

How is a directional derivative calculated?

The directional derivative is calculated using the gradient of the function and the direction vector. It is given by the dot product of the gradient vector and the unit vector in the direction of interest.

What is the significance of the directional derivative?

The directional derivative helps us understand the slope of a function in a specific direction. It is useful in optimization problems, where we want to find the direction in which the function changes the most.

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

The directional derivative has many applications in fields such as physics, engineering, and economics. It is used to study the behavior of physical systems, optimize designs, and analyze the effects of different inputs on a system.

What is the relationship between the directional derivative and the partial derivative?

The directional derivative is a generalization of the partial derivative. While the partial derivative measures the rate of change of a function with respect to a single variable, the directional derivative measures the rate of change of a function in a specific direction, which can involve multiple variables.

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