Unit Normal Vector: Surface S At Point M

by TextBrain Team 41 views

Alright, guys, let's dive into how to find the unit normal vector to a surface S at a specific point M. This is a common problem in multivariable calculus and vector calculus, and understanding the process is super useful for various applications in physics and engineering. We'll break it down step by step, making sure it's crystal clear. In our case, we have the surface S defined by the equation x² + xy² – 2z = 0 and the point M(1, 1, 1).

Understanding the Problem

Before we jump into the calculations, let's make sure we understand what we're trying to find. The normal vector to a surface at a given point is a vector that is perpendicular to the tangent plane of the surface at that point. Think of it as a line sticking straight out of the surface. A unit normal vector is simply a normal vector that has a length of 1. This normalization makes it easier to work with in many calculations.

The general approach involves these key steps:

  1. Define the Surface as a Level Surface: Rewrite the equation of the surface in the form F(x, y, z) = 0. This form is essential because it allows us to use the gradient to find the normal vector.
  2. Compute the Gradient: Calculate the gradient of F, denoted as ∇F. The gradient is a vector of partial derivatives, and it points in the direction of the greatest rate of increase of the function F. The gradient at a point on the surface is normal to the surface at that point.
  3. Evaluate the Gradient at the Point: Plug the coordinates of the point M into the gradient vector ∇F to find the specific normal vector at that point.
  4. Normalize the Vector: Divide the normal vector by its magnitude to obtain the unit normal vector. This ensures the vector has a length of 1.

Now, let's apply these steps to our specific problem.

Step 1: Define the Surface as a Level Surface

We're already given the surface in the form F(x, y, z) = 0:

F(x, y, z) = x² + xy² – 2z = 0

This is perfect! We can move straight to the next step.

Step 2: Compute the Gradient

The gradient of F is a vector of its partial derivatives with respect to x, y, and z:

∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z)

Let's calculate each partial derivative:

  • ∂F/∂x = 2x + y²
  • ∂F/∂y = 2xy
  • ∂F/∂z = -2

So, the gradient vector is:

∇F = (2x + y², 2xy, -2)

Step 3: Evaluate the Gradient at the Point M(1, 1, 1)

Now, we plug in the coordinates of the point M(1, 1, 1) into the gradient vector:

∇F(1, 1, 1) = (2(1) + (1)², 2(1)(1), -2) = (3, 2, -2)

This vector (3, 2, -2) is a normal vector to the surface S at the point M. However, it's not a unit vector yet. We need to normalize it.

Step 4: Normalize the Vector

To normalize the vector (3, 2, -2), we need to divide it by its magnitude. The magnitude of a vector (a, b, c) is given by:

||(a, b, c)|| = √(a² + b² + c²)

So, the magnitude of our normal vector (3, 2, -2) is:

||(3, 2, -2)|| = √(3² + 2² + (-2)²) = √(9 + 4 + 4) = √17

Now, we divide each component of the normal vector by its magnitude to get the unit normal vector:

Unit Normal Vector = (3/√17, 2/√17, -2/√17)

So, there you have it! The unit normal vector to the surface S at the point M is (3/√17, 2/√17, -2/√17).

Alternative Approach: Using Vector Functions

Another way to tackle this problem, especially if the surface is parameterized, is to use vector functions. Suppose the surface S is given by a vector function r(u, v), where u and v are parameters. Then, the normal vector to the surface is given by the cross product of the partial derivatives of r with respect to u and v:

N = ∂r/∂u × ∂r/∂v

Once you have the normal vector N, you can normalize it to find the unit normal vector, just like we did in the previous method. While this method isn't directly applicable to our given problem (since our surface is defined by an implicit equation), it's a useful technique to keep in your toolbox.

Common Mistakes to Avoid

  • Forgetting to Normalize: The most common mistake is finding the normal vector but forgetting to normalize it to get the unit normal vector. Always remember to divide by the magnitude!
  • Incorrect Partial Derivatives: Make sure you calculate the partial derivatives correctly. Double-check your work, especially with more complicated equations.
  • Sign Errors: Pay attention to signs when calculating the gradient and the magnitude. A small sign error can throw off the entire result.
  • Confusing the Order of Cross Product: If you're using the vector function approach, remember that the order of the cross product matters. ∂r/∂u × ∂r/∂v is not the same as ∂r/∂v × ∂r/∂u. The latter will give you a normal vector pointing in the opposite direction.

Importance of Understanding Normal Vectors

Understanding and being able to compute normal vectors is crucial in several fields:

  • Computer Graphics: Normal vectors are used for shading and lighting calculations. They determine how light reflects off a surface, which is essential for creating realistic images.
  • Physics: Normal vectors are used to calculate forces acting on surfaces, such as pressure and friction.
  • Engineering: Normal vectors are used in structural analysis to determine the stress and strain on surfaces.
  • Robotics: Normal vectors are used for collision detection and path planning.

Conclusion

Finding the unit normal vector to a surface at a given point is a fundamental skill in calculus and has wide-ranging applications. By following the steps outlined above – defining the surface as a level surface, computing the gradient, evaluating the gradient at the point, and normalizing the vector – you can confidently solve these types of problems. Remember to double-check your calculations and avoid common mistakes. Keep practicing, and you'll become a pro in no time! So next time, when someone asks, "Hey, can you find the unit normal vector?", you can confidently say, "Absolutely!"