Let a be a constant vector and 𝐫=x 𝐢+y 𝐣+z 𝐤. Verify the given identity. ∇·[(𝐫 ·𝐫) 𝐚]=2(𝐫 ·𝐚)

Name: Problem 18, Chapter 15: Calculus, Early Transcendentals
Uploaded: 2022-01-01T09:52:05-08:00
Duration: 1 min 14 s
Channel: Hast Aggarwal
Description: Let a be a constant vector and 𝐫=x 𝐢+y 𝐣+z 𝐤. Verify the given identity. ∇·[(𝐫 ·𝐫) 𝐚]=2(𝐫 ·𝐚)

step 1 So the question says that A be a constant vector, a constant vector, constant vector, A be a con

Let a be a constant vector and 𝐫=x 𝐢+y 𝐣+z 𝐤. Verify the...

Let a be a constant vector and $\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$. Verify the given identity.
$$
\nabla \cdot[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}]=2(\mathbf{r} \cdot \mathbf{a})
$$

Key Concepts

Constant Vectors

Constant vectors are vectors whose components do not depend on the spatial coordinates, meaning their derivatives with respect to any variable are zero. This property simplifies many vector calculus operations because when such a vector is involved, only the derivatives of the accompanying variable functions need to be considered.

Product Rule in Vector Calculus

The product rule in vector calculus extends the familiar rule from single-variable calculus to the differentiation of products involving scalar functions and vector fields. When a scalar function multiplies a vector field, the divergence of the product involves the gradient of the scalar and the divergence of the vector field, highlighting the interplay between scalar and vector quantities.

Dot Product

The dot product (or scalar product) is an operation that takes two vectors and produces a scalar by multiplying corresponding components and summing the results. It quantifies the projection of one vector onto another and is fundamental in determining quantities like work, energy, and in simplifying expressions in vector calculus.

Divergence Operator

The divergence operator measures the net rate at which flux is exiting an infinitesimally small volume around a point. In Cartesian coordinates, it is computed by summing the partial derivatives of each component of a vector field. This operator is essential for analyzing fields in physics and engineering, particularly in contexts like fluid flow and electromagnetic theory.

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