Problem 33 - Advanced Engineering Mathematics

Question

Show that $$\nabla \cdot \nabla f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}$$ This is known as the Laplacian and is also written \(\nabla^{2} f\).

Step-by-Step Solution

Answer

The Laplacian \( \nabla^2 f = \nabla \cdot \nabla f \) equals \( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \).

Step 1: Understanding the Problem

We need to demonstrate that the operator \( abla \cdot abla f \) is equivalent to the expression \( \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}} \), which represents the Laplacian of a function \( f \).

Step 2: Identifying the Gradient

The gradient of a scalar function \( f \) is given by the vector \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \). This vector represents the rate and direction of change of the function \( f \) at a point.

Step 3: Computing the Divergence

The divergence of a vector \( \mathbf{A} = (A_x, A_y, A_z) \) is \( abla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \). Here, we apply the divergence operator to \( abla f \), the gradient of the function.

Step 4: Applying Divergence to Gradient

Substituting \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \) into the divergence formula, we get:\[ abla \cdot abla f = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) + \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) + \frac{\partial}{\partial z}\left(\frac{\partial f}{\partial z}\right) \] This simplifies to:\[ abla \cdot abla f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \]

Step 5: Conclusion

We have shown that \( abla \cdot abla f = abla^2 f \), confirming it equals \( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \). Therefore, the statement is proven true, and the operator \( abla^2 \) is called the Laplacian.

Key Concepts

Gradient

The gradient is an essential concept in multivariable calculus. It is a vector that describes the direction and rate of fastest increase of a scalar function. Imagine standing on a surface representing a hill; the gradient tells you which direction to walk to increase your elevation the quickest.

Mathematically, if you have a function of three variables, say \( f(x, y, z) \), the gradient is represented as a vector:

• \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \)

The components of this vector are the partial derivatives of the function with respect to each variable. Each partial derivative indicates how the function changes if that particular variable is increased while the others are held constant.

To visualize, if the partial derivative \( \frac{\partial f}{\partial x} \) is high, then changing \( x \) will substantially increase \( f \). Thus, the gradient points towards the direction of steepest ascent on a surface, and its magnitude tells how steep that ascent is.

Divergence

Divergence measures the magnitude of a source or sink at a given point in a vector field. Essentially, it tells us how much "stuff" is expanding or contracting from that point.

In practical terms, for a vector field \( \mathbf{A} = (A_x, A_y, A_z) \) in three-dimensional space, the divergence is defined as:

• \( abla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \)

The divergence at a point reflects whether vectors at that location are spreading out (positive divergence) or converging in (negative divergence).

In the context of our exercise, when the divergence operator is applied to a gradient, it results in the Laplacian, linking these concepts by showing how the function expands or contracts around that point.

Partial Derivatives

Partial derivatives are fundamental in describing how a multi-variable function changes with respect to one variable at a time, keeping others constant.

If \( f(x, y, z) \) is a function of three variables, then its partial derivatives are:

• \( \frac{\partial f}{\partial x} \) - how the function changes with a tiny change in \( x \), keeping \( y \) and \( z \) constant.
• \( \frac{\partial f}{\partial y} \) - change in \( y \).
• \( \frac{\partial f}{\partial z} \) - change in \( z \).

For second order partial derivatives, denoted as \( \frac{\partial^2 f}{\partial x^2} \), we observe how the rate of change itself changes, further diving into the function's behavior.

These derivatives give us crucial insights about curvature and concavity of the surface described by the function, forming the core in calculation of the Laplacian, as they express how the function evolves around a specific point in space.