Question

Looking ahead- tangent planes Consider the following surfaces \(f(x, y, z)=0,\) which may be regarded as a level surface of the function \(w=f(x, y, z) .\) A point \(P(a, b, c)\) on the surface is also given. a. Find the (three-dimensional) gradient of \(f\) and evaluate it at \(P\). b. The set of all vectors orthogonal to the gradient with their tails at \(P\) form a plane. Find an equation of that plane (soon to be called the tangent plane). $$f(x, y, z)=x^{2}+y^{2}+z^{2}-3=0 ; P(1,1,1)$$

Short answer

Question: Find the tangent plane of the level surface represented by the function \(f(x, y, z) = x^2 + y^2 + z^2 - 3\) at the point P(1, 1, 1). Answer: The tangent plane at point P(1, 1, 1) is given by the equation \(x + y + z = 3\).

Step-by-step solution

Part a: Find the Gradient

To find the gradient of \(f(x, y, z)\), we need to compute the partial derivatives of the function with respect to x, y, and z: $$\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$$ Now let's compute the partial derivatives: $$\frac{\partial f}{\partial x} = 2x$$ $$\frac{\partial f}{\partial y} = 2y$$ $$\frac{\partial f}{\partial z} = 2z$$ Thus, the gradient of \(f\) is: $$\nabla f = 2x\mathbf{i} + 2y\mathbf{j} + 2z\mathbf{k}$$ Now, let's evaluate the gradient at the point P(1, 1, 1): $$\nabla f_P = 2(1)\mathbf{i} + 2(1)\mathbf{j} + 2(1)\mathbf{k} = 2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$$

Part b: Find the Tangent Plane Equation

Now that we have the gradient at point P, we can find the equation of the tangent plane using the fact that the gradient vector is normal to the tangent plane. The equation of the tangent plane is given by: $$\nabla f_P \cdot (x - x_P, y - y_P, z - z_P) = 0$$ Plugging in the values, we get: $$(2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}) \cdot (x - 1, y - 1, z - 1) = 0$$ $$2(x - 1) + 2(y - 1) + 2(z - 1) = 0$$ Finally, simplifying the equation, we get the equation of the tangent plane at point P(1, 1, 1): $$x + y + z = 3$$

Key concepts

Gradient

The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function. This is especially useful for understanding surfaces described by equations like \( f(x, y, z) = 0 \).
Here, a gradient \( abla f \) is a vector composed of the partial derivatives of \( f \) with respect to each of its variables, which are \( x, y, \) and \( z \).
The explicit formula for the gradient is:
\[ abla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} \]
In the example provided, the gradient becomes \( 2x\mathbf{i} + 2y\mathbf{j} + 2z\mathbf{k} \), showing how each directional derivative contributes to the overall vector.

• The gradient tells us not just the direction, but its magnitude also provides the rate of change.
• At any point \( P(a, b, c) \), evaluating the gradient gives specifics about the behavior of the surface at that point.

The gradient is crucial in finding vectors normal to the tangent plane, as it always points perpendicular to the level surface.

Partial Derivatives

Partial derivatives are fundamental in multivariable calculus. They measure how a function changes as each variable changes, while other variables remain fixed.
For a function \( f(x, y, z) \), partial derivatives with respect to \( x, y, \) and \( z \) are denoted by:
\[ \frac{\partial f}{\partial x} \; \text{for changes in} \; x \]
\[ \frac{\partial f}{\partial y} \; \text{for changes in} \; y \]
\[ \frac{\partial f}{\partial z} \; \text{for changes in} \; z \]
In the example:

• \( \frac{\partial f}{\partial x} = 2x \): only \( x \) changes and contributes to this derivative.
• \( \frac{\partial f}{\partial y} = 2y \): similarly, change in \( y \) affects this one.
• \( \frac{\partial f}{\partial z} = 2z \): change in \( z \) determines this rate.

Each partial derivative tells us how the function rises or falls as each individual input increases while holding the other inputs constant. Calculating these is the first step in constructing the entire gradient vector.

Level Surface

A level surface of a function \( w = f(x, y, z) \) is where the function remains constant at a specific value, often \( w = 0 \).
This means any point \( (x, y, z) \) on this surface satisfies the equation precisely.

In the given example, the level surface is defined by \( x^2 + y^2 + z^2 - 3 = 0 \). Such an equation forms a geometric region in space, like a sphere, where \( x, y, \) and \( z \) values result in the same function value.

• Level surfaces are essentially the 3D analogs of level curves (or contour lines) in 2D.
• They help visualize where the function value is the same in all directions perpendicular to the gradient.

By understanding level surfaces, we can analyze the geometry of a surface, its orientation, and properties of tangents, helping in visualizing complex function behaviors in 3D.