# Remarks by Fortney Following Theorems on Directional Derivative ....

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In summary, Fortney discusses the equation for a plane through a point on a graph and explains that it can be derived by translating the origin to that point. This is done by replacing the variables with their differences from the point's coordinates.
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MHB
I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ...I need help to understand some remarks by Fortney following Theorems 2.1 and 2.2 on the directional derivative ...

Those remarks by Fortney read as follows: (for more text showing the context of the remarks including the two Theorems ... see scanned text below ...)View attachment 8776
In the above text by Fortney we read the following:

" ... ... From this it is straightforward to show that the equation of a plane through the point $$\displaystyle (x_0, y_0, f( x_0, y_0) )$$ on the graph of $$\displaystyle f$$ is given by

$$\displaystyle z - f( x_0, y_0) = m_x ( x - x_0 ) + m_y ( y - y_0 )$$
... ... "Can someone please explain/demonstrate exactly how the above equation arises or comes about ...?
Help will be appreciated ...

Peter========================================================================================So as to make clear the context of the above qestion I am providing Fortney's text before and after the text provided above ... as follows:View attachment 8777
View attachment 8776
View attachment 8778Hope that helps ...Peter

#### Attachments

• Fortney - 2 - Remarks following Theorems 2.1 and 2.2 ... PART 2 .png
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• Fortney - 2 - Remarks following Theorems 2.1 and 2.2 ... PART 2 .png
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• Fortney - 1 - Remarks following Theorems 2.1 and 2.2 ... PART 1 .png
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• Fortney - 3 - Remarks following Theorems 2.1 and 2.2 ... PART 3 .png
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Peter said:
In the above text by Fortney we read the following:

" ... ... From this it is straightforward to show that the equation of a plane through the point $$\displaystyle (x_0, y_0, f( x_0, y_0) )$$ on the graph of $$\displaystyle f$$ is given by

$$\displaystyle z - f( x_0, y_0) = m_x ( x - x_0 ) + m_y ( y - y_0 )$$
... ... "Can someone please explain/demonstrate exactly how the above equation arises or comes about ...?

Do you already understand where the equation given for a plane through the origin comes from? If so, it's just a translation!
Fortney gives that the equation for a plane through the origin is...
$z = m_xx + m_yy$
Note that what we have here is...
$(z-z_0) = m_x(x-x_0) + m_y(y-y_0)$ where $(x_0, y_0, z_0) = (0, 0, 0)$ (the origin).

So in order to get a plane through our point $(x_0, y_0, f(x_0, y_0))$, we can translate the origin to this point by simply replacing $x$ with $x-x_0$, $y$ with $y-y_0$, and $z$ with $z-z_0$, giving us
$z - f(x_0, y_0) = m_x(x-x_0) + m_y(y-y_0)$

joypav said:
Do you already understand where the equation given for a plane through the origin comes from? If so, it's just a translation!
Fortney gives that the equation for a plane through the origin is...
$z = m_xx + m_yy$
Note that what we have here is...
$(z-z_0) = m_x(x-x_0) + m_y(y-y_0)$ where $(x_0, y_0, z_0) = (0, 0, 0)$ (the origin).

So in order to get a plane through our point $(x_0, y_0, f(x_0, y_0))$, we can translate the origin to this point by simply replacing $x$ with $x-x_0$, $y$ with $y-y_0$, and $z$ with $z-z_0$, giving us
$z - f(x_0, y_0) = m_x(x-x_0) + m_y(y-y_0)$
Thanks joypav ...

Yes ... follow that ... straightforward when you see how ...

Peter

## 1. What are directional derivatives?

Directional derivatives are a type of derivative used in multivariable calculus that measures the rate of change of a function in a specific direction. It is represented by the symbol ∇f and is calculated using partial derivatives.

## 2. What are the theorems related to directional derivatives?

The two main theorems related to directional derivatives are the Chain Rule and the Directional Derivative Theorem. The Chain Rule allows us to calculate directional derivatives of composite functions, while the Directional Derivative Theorem states that the directional derivative of a function at a point is equal to the dot product of the gradient of the function at that point and the unit vector in the direction of interest.

## 3. How are directional derivatives used in real-world applications?

Directional derivatives have many real-world applications, especially in fields such as physics, engineering, and economics. They can be used to analyze the rate of change of physical quantities, such as velocity and acceleration, and to optimize functions in various industries, such as maximizing profits in business.

## 4. Can directional derivatives be negative?

Yes, directional derivatives can be negative. This indicates that the function is decreasing in the direction of interest. A positive directional derivative indicates that the function is increasing in that direction, and a value of zero means that the function is constant in that direction.

## 5. How do I calculate directional derivatives?

To calculate a directional derivative, you need to first find the gradient of the function at the point of interest. Then, you multiply the gradient vector by the unit vector in the direction you want to find the derivative in. This will give you the directional derivative at that point.

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