- #1

- 82

- 0

Hi all,

According to wikipedia:

Can someone explain to me with a mathematical proof the following:

$$ \frac {\partial f(x)} {\partial v} = \hat v \cdot \nabla f(x) $$

I don't get this identity except the special example where the partial derivative of f(x) wrt x is a special kind of a directional derivative along the x axis, because the other components of the gradient vector cancel in the dot product with the unit vector along the direction of x.

$$ \frac {\partial f(x)} {\partial x} = \hat i \cdot \nabla f(x) $$

So can someone explain to me in a mathematical way how to proof the general identity?

Identity:

$$ \frac{\partial f(x)}{\partial v} = \hat v . \nabla f(x) $$

So gradient is:

$$ \nabla f(x)= \frac{\partial f(x)}{\partial x} \hat i +\frac{\partial f(x)}{\partial y} \hat j +\frac{\partial f(x)}{\partial z} \hat k $$

Random unit vector is:

$$ \hat v = v_x \hat i + v_y \hat j+ v_z \hat k $$

RHS of identity:

$$\hat v . \nabla f(x)= v_x \frac{\partial f(x)}{\partial x} + v_y \frac{\partial f(x) }{\partial y} + v_z \frac{\partial\ f(x) }{\partial z} $$

LHS:

$$ \frac{\partial f(x)}{\partial v} = v_x \frac{\partial f(x)}{\partial x} + v_y \frac{\partial f(x) }{\partial y} + v_z \frac{\partial\ f(x) }{\partial z} $$

I don't see it how the partial derivative on the left side is actually a directional derivative.

In plain english: where does the v comes from in $$ \frac{\partial f(x)}{\partial v} $$ ?

According to wikipedia:

Directional derivatives can be also denoted by:

wherevis a parameterization of a curve to whichvis tangent and which determines its magnitude.

Can someone explain to me with a mathematical proof the following:

$$ \frac {\partial f(x)} {\partial v} = \hat v \cdot \nabla f(x) $$

I don't get this identity except the special example where the partial derivative of f(x) wrt x is a special kind of a directional derivative along the x axis, because the other components of the gradient vector cancel in the dot product with the unit vector along the direction of x.

$$ \frac {\partial f(x)} {\partial x} = \hat i \cdot \nabla f(x) $$

So can someone explain to me in a mathematical way how to proof the general identity?

Identity:

$$ \frac{\partial f(x)}{\partial v} = \hat v . \nabla f(x) $$

So gradient is:

$$ \nabla f(x)= \frac{\partial f(x)}{\partial x} \hat i +\frac{\partial f(x)}{\partial y} \hat j +\frac{\partial f(x)}{\partial z} \hat k $$

Random unit vector is:

$$ \hat v = v_x \hat i + v_y \hat j+ v_z \hat k $$

RHS of identity:

$$\hat v . \nabla f(x)= v_x \frac{\partial f(x)}{\partial x} + v_y \frac{\partial f(x) }{\partial y} + v_z \frac{\partial\ f(x) }{\partial z} $$

LHS:

$$ \frac{\partial f(x)}{\partial v} = v_x \frac{\partial f(x)}{\partial x} + v_y \frac{\partial f(x) }{\partial y} + v_z \frac{\partial\ f(x) }{\partial z} $$

I don't see it how the partial derivative on the left side is actually a directional derivative.

In plain english: where does the v comes from in $$ \frac{\partial f(x)}{\partial v} $$ ?

Last edited: