91Ó°ÊÓ

In the realm of multivariable calculus, the gradient is a crucial concept. It consists of partial derivatives of a multivariable function, bundling them into a vector. Consider a function like \( f(x, y) = xy \). By taking the partial derivatives with respect to \( x \) and \( y \), we find the gradient \( abla f \).
The partial derivative \( \frac{\partial f}{\partial x} \) measures how \( f \) changes as \( x \) changes, with \( y \) held constant, and vice versa.
For our exercise, substituting \( y = 3 \), we compute the gradient:

• \( \frac{\partial f}{\partial x} = 3 \)
• \( \frac{\partial f}{\partial y} = t^2 + 1 \)

The gradient vector \( abla f = (3, t^2 + 1) \) points in the direction of the steepest ascent of \( f \).

The velocity vector is a fundamental concept in physics and calculus that represents the rate of change of a position vector with respect to time. This vector shows how an object's position changes over time.
To find the velocity vector \( \mathbf{v} \), differentiate the position vector \( \mathbf{r}(t) = (x(t), y(t)) \) with respect to time.

• Given \( x(t) = t^2 + 1 \), we find \( \dot{x}(t) = 2t \).
• Since \( y(t) = 3 \) is constant, \( \dot{y}(t) = 0 \).

This yields the velocity vector \( \mathbf{v} = (2t, 0) \). It tells us the object moves horizontally with a velocity proportional to \( t \).

The tangent vector provides the direction and rate at which a curve progresses at a particular point. It is aligned along the path of the curve, showing the trajectory of movement. The tangent vector \( \mathbf{T} \) can be derived from the velocity vector \( \mathbf{v} \) by normalization.
Normalization ensures that \( \mathbf{T} \) is a unit vector, which preserves the direction of \( \mathbf{v} \) but sets its length to 1.
To normalize:

• Calculate the magnitude of \( \mathbf{v} \): \( ||\mathbf{v}|| = \sqrt{(2t)^2 + 0^2} = 2|t| \).
• Normalize \( \mathbf{v} \) to get \( \mathbf{T} = \left(\frac{2t}{2|t|}, 0\right) = (\text{sgn}(t), 0) \).

The resulting unit vector \( \mathbf{T} \) simply points in the direction of \( \mathbf{v} \) and indicates the path tangent to the object's course.

The rate of change in calculus is an essential concept that quantifies how a quantity alters over time or along a path. In our context, we're examining two specific rate of changes:
1. **With respect to time (\( \frac{df}{dt} \))**: This reflects how the function \( f \) changes as time progresses. It's calculated as the dot product of the gradient \( abla f \) and the velocity vector \( \mathbf{v} \). For our function, this results in \( \frac{df}{dt} = abla f \cdot \mathbf{v} = 6t \). This signifies that the rate of change of \( f \) over time is proportional to \( t \).
2. **With respect to space (\( \frac{df}{ds} \))**: Here, we look at how \( f \) changes along a specific path, parallel to the tangent vector \( \mathbf{T} \). Again, using a dot product, \( \frac{df}{ds} = abla f \cdot \mathbf{T} = 3 \cdot \text{sgn}(t) \). This provides the slope, which describes \( f \)'s steepness along the curve.
By understanding these rates, one can discern how \( f \) evolves over time and space, providing insight into both motion and change.