Let C be the curve which is the intersection of the half-cone S1 = {(x,y,z)|z=sqrt(x^2 + y^2} and the paraboloid S2 = {(x,y,z)|2z=3-x^2-y^2}. Find C.
a)Make a 3-D sketch to show S1, S2, and C
b) Show that at each point on C the normals to these two surfaces are perpendicular to each other.
To find the curve C, which is the intersection of the half-cone S1 and the paraboloid S2, we can follow these steps:
a) 3-D Sketch:
To make a 3-D sketch, it's helpful to visualize each surface separately and then find their intersection. Here are the steps to sketch S1, S2, and C:
1. Sketch S1 (half-cone):
- Start with a 3-D coordinate system.
- The equation of S1 is z = sqrt(x^2 + y^2), which represents a cone opening upwards.
- For various values of x and y, calculate z using the equation.
- Plot these points and connect them to form the surface S1.
2. Sketch S2 (paraboloid):
- The equation of S2 is 2z = 3 - x^2 - y^2, which represents an upside-down bowl-shaped surface.
- For different values of x and y, calculate z using the equation.
- Plot these points and connect them to form the surface S2.
3. Find the Intersection Curve C:
- Locate the points where S1 and S2 intersect.
- These points should satisfy both the equation of S1 and S2 simultaneously.
- Plot these points, and they will form the curve C.
b) Perpendicular Normals:
To show that at each point on C, the normals to S1 and S2 are perpendicular to each other, we need to consider the gradient vectors of S1 and S2 at those points.
1. Calculate the gradient vector of S1:
- The gradient vector of S1 is given by (∂z/∂x, ∂z/∂y, -1).
- Compute the partial derivatives ∂z/∂x and ∂z/∂y by differentiating the equation z = sqrt(x^2 + y^2) with respect to x and y, respectively.
- The negative sign in the z-component is because the surface S1 opens upwards.
2. Calculate the gradient vector of S2:
- The gradient vector of S2 is given by (∂z/∂x, ∂z/∂y, -2).
- Compute the partial derivatives ∂z/∂x and ∂z/∂y by differentiating the equation 2z = 3 - x^2 - y^2 with respect to x and y, respectively.
- The negative sign in the z-component is because the surface S2 is upside-down.
3. Evaluate the gradient vectors at each point on C:
- Take any point on the curve C, and calculate the gradient vector of S1 and S2 at that point.
- Dot product the two gradient vectors. If the dot product is zero, then the normals are perpendicular.
By following these steps, you can sketch S1, S2, and find the curve C, as well as show that the normals to S1 and S2 are perpendicular at each point on C.