- #1
brendan
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Homework Statement
Describe the level surface
Homework Equations
f(x,y,z) = 3x - y + 2z
The Attempt at a Solution
f(x,y,z) = 3x - y + 2z
3x - y + 2z = w
f(-1,1,2) for w <= 0 <= is a plane
regards
Brendan
?f(-1,1,2) for w <= 0 <= is a plane
has no real meaning, f(-1,1,2) represents a scalar value and I'm not sure what you mean by your comaprison sybols for wbrendan said:f(-1,1,2) for w <= 0 <= is a plane
is close to the moneybrendan said:is a plane
brendan said:Thanks about the method of graphing the function it helped alot!
Would you say that when
"describing the function f(x, y, z), it's probably sufficient to say that the level surfaces are all planes."
would you mention the magnitude of the plan eg
The level surfaces of the function f(x, y, z) = w are all planes at w/2
regards
Brendan
Each level surface is a plane.brendan said:So the level surface represent parallel planes.
For f(x, y, z) = 3x -y + 2z, the level surface f(x, y, z) = 0 is a plane that contains the three points below.brendan said:For f(x,y,z) 3x - y + 2z = 0
brendan said:We give three points
P1(0,3,1.5)
P2(0,1,1/2)
P3(0,2,1)
All are point of a unique plane.
With Normal vector
P1->P2(0,-2,-1)
P1->P3(0,-1,-1/2)
P1->P2 X P1->P3
normal vector (0,0,0)
I'm not sure why you are focusing so much on w= 0. It should be immediately clear, from what you have already learned about planes, that f(x,y,z)= 3x - y + 2z = Constant is a plane with normal vector (3, -1, 2) for any Constant.brendan said:So the level surface represent parallel planes.
For f(x,y,z) 3x - y + 2z = 0
We give three points
P1(0,3,1.5)
P2(0,1,1/2)
P3(0,2,1)
All are point of a unique plane.
With Normal vector
P1->P2(0,-2,-1)
P1->P3(0,-1,-1/2)
P1->P2 X P1->P3
normal vector (0,0,0)
A level surface is a two-dimensional surface in three-dimensional space where every point on the surface has the same value for a given function. This means that if you were to graph the function on the surface, all the points would be at the same height.
A contour line is a one-dimensional curve on a two-dimensional surface that connects points of equal value on a function. In contrast, a level surface is a two-dimensional surface that contains all the points of equal value for a function in three-dimensional space.
The equation for a level surface is given by f(x, y, z) = k, where f is a function of three variables and k is a constant value. This equation represents all the points on the surface where the function has a value of k.
Level surfaces are useful in many scientific fields, such as physics, engineering, and geography. They can help visualize and analyze three-dimensional functions and their properties, such as slope, curvature, and critical points.
No, level surfaces can have different shapes and curvatures depending on the function and the constant value. Some level surfaces may be flat, while others may be curved or even irregular.