derivative of vector norm
Understand the Problem
The question is asking for the mathematical derivative of the norm of a vector, which typically involves applying calculus to understand how the vector norm changes with respect to its components.
Answer
The derivative of the vector norm with respect to $v_j$ is $\frac{d}{dv_j} \|\mathbf{v}\| = \frac{v_j}{\|\mathbf{v}\|}$.
Answer for screen readers
The derivative of the norm of a vector $\mathbf{v}$ with respect to its component $v_j$ is given by: $$ \frac{d}{dv_j} |\mathbf{v}| = \frac{v_j}{|\mathbf{v}|} $$
Steps to Solve
-
Identify the norm of the vector For a vector $\mathbf{v} = (v_1, v_2, \ldots, v_n)$, the norm (or magnitude) is given by the formula: $$ |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + \ldots + v_n^2} $$
-
Apply the chain rule for differentiation To find the derivative of the norm with respect to one of the vector components, we apply the chain rule. The derivative of $|\mathbf{v}|$ with respect to $v_j$ is: $$ \frac{d}{dv_j} |\mathbf{v}| = \frac{1}{2|\mathbf{v}|} \frac{d}{dv_j} (v_1^2 + v_2^2 + \ldots + v_n^2) $$
-
Differentiate the squared terms Differentiate the sum of the squares: $$ \frac{d}{dv_j} (v_1^2 + v_2^2 + \ldots + v_n^2) = 2v_j $$
-
Combine the results Now substitute the derivative back into the equation from step 2: $$ \frac{d}{dv_j} |\mathbf{v}| = \frac{1}{2|\mathbf{v}|} \cdot 2v_j = \frac{v_j}{|\mathbf{v}|} $$
-
Final result Thus, the derivative of the norm of the vector $\mathbf{v}$ with respect to the component $v_j$ is: $$ \frac{d}{dv_j} |\mathbf{v}| = \frac{v_j}{|\mathbf{v}|} $$
The derivative of the norm of a vector $\mathbf{v}$ with respect to its component $v_j$ is given by: $$ \frac{d}{dv_j} |\mathbf{v}| = \frac{v_j}{|\mathbf{v}|} $$
More Information
In vector calculus, the derivative of the norm provides important information about how the length of the vector changes as its components change. This concept is often utilized in optimization problems and physics, especially in understanding directions of steepest ascent or descent.
Tips
- Forgetting the chain rule: It’s crucial to remember the chain rule when differentiating composite functions like the norm.
- Neglecting the vector's magnitude: Ensure that you don’t confuse the derivative of the vector components with the derivative of its norm.
