To solve the problem, we need to identify which of the given expressions is not equivalent to the others. The expressions involve vector dot products and time derivatives.

Analyzing the Expressions:

1. Expression 1: $\frac{1}{2} \left[ \dot{a}_1 b_2 + \dot{a}_2 b_2 + \dot{a}_3 b_3 - a_1 \dot{b}_1 - a_2 \dot{b}_2 - a_3 \dot{b}_3 \right]$

2. Expression 2: $\dot{\mathbf{a}} \cdot \mathbf{b} + \mathbf{a} \cdot \dot{\mathbf{b}}$ This is the product rule for the time derivative of the dot product of two vectors $\mathbf{a}$ and $\mathbf{b}$: $\frac{d}{dt} (\mathbf{a} \cdot \mathbf{b}) = \dot{\mathbf{a}} \cdot \mathbf{b} + \mathbf{a} \cdot \dot{\mathbf{b}}$

3. Expression 3: $\frac{d}{dt} (\mathbf{a} \cdot \mathbf{b})$ This is exactly the time derivative of the dot product $\mathbf{a} \cdot \mathbf{b}$. Using the product rule, it can be expanded to: $\frac{d}{dt} (\mathbf{a} \cdot \mathbf{b}) = \dot{\mathbf{a}} \cdot \mathbf{b} + \mathbf{a} \cdot \dot{\mathbf{b}}$

4. Expression 4: $\dot{a}_1 b_1 + \dot{a}_2 b_2 + \dot{a}_3 b_3 + a_1 \dot{b}_1 + a_2 \dot{b}_2 + a_3 \dot{b}_3$ This is another form of expanding the product rule result. It looks like the detailed expansion of Expression 2.

Comparison:

• Expression 1 has some negative terms and positive terms which are different from the straightforward application of the product rule.
• Expressions 2, 3, and 4 are variations or direct consequences of the product rule.

Conclusion:

Expression 1 is not equivalent to the others, because it does not match the result given by the product rule. The sum of derivatives in Expression 1 is incorrectly constructed with some signs opposite to what they should be.

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. How is the product rule applied to vector dot products?
2. Can you derive the product rule for a general vector function?
3. What are some other common mistakes when applying the product rule in vector calculus?
4. How does the product rule differ for cross products?
5. What is the physical interpretation of the product rule in vector calculus?

Tip: Always check the signs and terms carefully when applying rules like the product rule in vector calculus, as small errors can lead to incorrect conclusions.