The discussion centers on the mathematical problem of demonstrating that if the gradient of a function \( f(x,y,z) \) is always parallel to the position vector \( \mathbf{r} = xi + yj + zk \), then it follows that \( f(0,0,a) = f(0,0,-a) \) for any value of \( a \). Participants explored the implications of the dot product between the gradient and the position vector, emphasizing the significance of the angle between them. The conclusion drawn is that the vector from \( f(0,0,a) \) to \( f(0,0,-a) \) is indeed \( -2ak \), and further analysis through unit vectors and dot products is necessary to validate the relationship.
PREREQUISITESThis discussion is beneficial for students and educators in mathematics, particularly those studying multivariable calculus, as well as anyone interested in the geometric interpretations of gradients and vector fields.
Mathgician said:dot product, or cosine of theta
positive if theta is 0, negative if theta is 180 or pi
that tells your max and min slope
oahsen said:what should ı do with the max and min slope value?
Mathgician said:what is the vector from f(0,0,a) to f(0,0,-a)?
find the unit vector of this vector,
and then do the dot product of the unit vector and the gradient vector.
oahsen said:since we do not know f how can we find the vector from f(0,0,a) to f(0,0,-a)?
please be more clear. I do not know perfectly this subject.
Mathgician said:lets call f(0,0,a) point P, and f(0,0,-a) point S
What is vector PS?