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The concept of divergence in vector calculus is crucial for understanding how a vector field behaves. It measures the magnitude of a source or sink at a given point. Consider the divergence operator \( abla \cdot \), also known as del dot, applied to a vector field. This operator essentially sums up all the outgoing vector field strengths from a point.

In the given problem, you deal with \( abla \cdot [(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}] \). Here, \(\mathbf{a}\) is a constant vector, and the expression represents the divergence of the product of a scalar and a vector. Usually, the formula for divergence of a product is \( abla \cdot (f \mathbf{a}) = abla f \cdot \mathbf{a} + f (abla \cdot \mathbf{a}) \). Since \(\mathbf{a}\) is constant, its divergence is zero, simplifying your task.

• Understand outlet intensity of vector fields at a point with divergence.
• Dive deep into expressions involving scalars and vectors.

The dot product, often called the scalar product, is fundamental in vector calculus and geometry. It offers a way to multiply two vectors, producing a scalar. For vectors \( \mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k} \) and \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \), the dot product is given by:

• \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \)

In the exercise, \( \mathbf{r} \cdot \mathbf{r} \) results in \( x^2 + y^2 + z^2 \), which is a scalar since \( \mathbf{r} \) is dotted with itself. Another example is \( 2(\mathbf{r} \cdot \mathbf{a}) \), simplifying the vector expression \( \mathbf{r} \cdot \mathbf{a} = xa_1 + ya_2 + za_3 \) due to multiplicity by two.

• Remember: dot product transforms vector pair into scalar value.
• Useful for determining angles and projections in space.

The gradient is an operator denoted by \( abla \), and it indicates the direction of the steepest ascent of a scalar field. Calculating the gradient of a scalar function \( f(x, y, z) \) involves taking the partial derivatives with respect to each coordinate:

• \( abla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} \)

In the problem, you compute \( abla (x^2 + y^2 + z^2) \), which results in \( 2x \mathbf{i} + 2y \mathbf{j} + 2z \mathbf{k} \). This gradient is significant because it represents the vector field created by the scalar function \( x^2 + y^2 + z^2 \).

• Visualize gradient as a vector pointing to highest increase rate in field.
• Provides information about field changes and directions.

Scalar multiplication in vector math involves scaling a vector by a scalar, affecting its magnitude but not its direction. Suppose you have a vector \( \mathbf{v} \) and a scalar \( c \), then multiplying results in a new vector \( c \mathbf{v} \). Each component of \( \mathbf{v} \) is multiplied by \( c \):

• If \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \), then \( c \mathbf{v} = c v_1 \mathbf{i} + c v_2 \mathbf{j} + c v_3 \mathbf{k} \)

Within the exercise, the expression \( (\mathbf{r} \cdot \mathbf{r}) \mathbf{a} \) invokes scalar multiplication. Here, \( \mathbf{r} \cdot \mathbf{r} = x^2 + y^2 + z^2 \) forms the scalar scaling the constant vector \( \mathbf{a} \). This simplifies the process of resolving vector intensities.

• Single-number multiplication making vector longer or shorter.
• Preserves direction unless the scalar is negative.