phyzguy said:I think you've calculated the gradient correctly, but not the normal unit vector. The gradient is tangent to the surface in the direction where the slope is maximum. The normal vector is perpendicular to the tangent plane of the surface. So the normal unit vector will be perpendicular to the gradient vector. You have just calculated the unit gradient vector.
ehild said:Tomanator,
your solution is correct. The gradient in a point of a f(x,y,z)=constant surface is normal to the tangent plane of this surface at the given point.
A small change of f can be written as dF=grad f˙dr, the dot product of the gradf vector with the displacement vector dr. You get the highest change of f if the displacement is parallel to the gradient: the gradient vector points in the direction of the steepest slope of the f function.
If dr is tangent to an f(x,y,z) = constant surface dF=grad f˙dr=0. A dot product of two non-zero vector is 0 if they are perpendicular: gradf and dr are perpendicular, gradf is normal vector to the tangent plane.
ehild
tomanator said:Homework Statement
For the scalar field f(x, y, z) = x2 − y2 − z find gradf and normal unit
vector to a surface f(x, y, z) = 0 at the point (1, 1, 0).
Homework Equations
The Attempt at a Solution
I calculated gradf= 2xi -2yj -k
at (1,1,0) this is = 2i -2y -k
normal unit vector= [tex]\frac{2i -2y -k}{\sqrt{9}}[/tex]
normal unit vector = [tex]\frac{1}{3}[/tex](2i -2y -k)
is this correct it seems too simple in terms of marks
msslowlearner said:yes, i get it .. i always keep forgetting dr is the tangent to the plane of surface. thanks ehild. could you help with what a unit normal vector is ?? is it the direction of the gradient vector? if so, how do you calculate it given the gradient and a point on the surface ?
Ray Vickson said:Your calculations are correct: grad f is the direction of steepest ascent (i.e., the direction of greatest rate of increase in f), and grad(f)/|grad(f)| is the corresponding unit normal.
RGV
A vector is a mathematical quantity that has both magnitude and direction. It is often represented as an arrow in a coordinate system.
A gradient vector is a vector that points in the direction of the steepest increase of a function at a specific point. It is perpendicular to the level curves of the function.
A gradient vector is calculated by taking the partial derivatives of a function with respect to each variable and combining them into a vector. The resulting vector represents the direction and magnitude of the steepest increase of the function.
A normal unit vector is a vector that is perpendicular to a surface at a specific point and has a magnitude of 1. It is often used in physics and engineering to represent forces or directions of motion.
Yes, it is correct to use a normal unit vector to represent a force. In physics, forces are often represented by vectors, and the normal unit vector is a commonly used vector to represent forces that are perpendicular to a surface. However, it is important to ensure that the direction and magnitude of the vector accurately represent the force being applied.