Show that \((\mathbf{A} \cdot \nabla) \mathbf{r}=\mathbf{A}\).

The gradient operator, often represented by the symbol \( abla \) — pronounced 'nabla' — is a cornerstone of vector calculus, especially useful in fields like physics and engineering. Understanding the gradient operator is crucial for anyone delving into the realms of vector calculus.

Imagine you are standing on a hill, the gradient operator would tell you the direction of the steepest slope from where you stand. Mathematically, in three-dimensional space, the gradient of a scalar function is a vector that points in the direction of the greatest rate of increase of the function and whose magnitude is the rate of increase in that direction.

The gradient operator in Cartesian coordinates is defined as \( abla = \big( \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z} \big) \), composed of partial derivatives with respect to each spatial variable. When applied to a scalar field, these components individually compute the rate of change along each axis of the space, forming a vector that represents the collective rate of change within the field.

The dot product, also known as the scalar product, is a fundamental operation in vector calculus that combines two vectors to produce a scalar value. This scalar value is immensely informative — it tells us about the magnitude of projection of one vector onto another and can also indicate the angle between them.

The dot product is calculated by multiplying the corresponding components of the two vectors and then summing those products. The formula for the dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) with components \( A_x, A_y, A_z \) and \( B_x, B_y, B_z \) respectively, is \( \mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y + A_zB_z \).

If the vectors are perpendicular, the dot product is zero — this is useful when checking for orthogonality. Additionally, if either of the vectors is zero or if the angle between them is 90 degrees, the dot product will also return as zero, reflecting this orthogonal or 'no projection' relationship.

The position vector is a fundamental concept in physics and engineering that describes the location of a point in space relative to an origin. Typically notated as \( \mathbf{r} \), it is an arrow drawn from the origin of the coordinate system to the point in question. In a three-dimensional space, the position vector can be expressed as \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), with \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) being the unit vectors in the x, y, and z directions respectively.

The concept of a position vector is crucial as it allows us to apply operations such as the gradient or the dot product to physical locations. These operations can reveal directional rates of change and projections, linking algebraic computations to real-world geometrical interpretations. Understanding how position vectors interact with other vectors and operators is foundational for visualizing and solving problems in vector calculus.