91Ó°ÊÓ

The gradient vector, often represented as \(abla f\), plays a crucial role in understanding how a function changes at any point on its surface. For a function \(f(x, y, z)\), the gradient is a vector consisting of the partial derivatives with respect to \(x\), \(y\), and \(z\). In simple terms, \(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)\).

• The components of \(abla f\) reveal the function's rate of increase in the direction of each coordinate axis.
• For our example surface \(f(x, y, z) = x^2 + y^2 - z\), the gradient becomes \((2x, 2y, -1)\).

To check if a curve is normal to a surface at a specific point, comparing the curve's velocity vector to the surface's gradient vector helps. If the velocity vector is a scalar multiple of the gradient vector, that signifies perpendicularity, indicating the surface is locally flat with respect to the curve at that point.

The velocity vector of a curve, such as \(\mathbf{v}(t)\), represents how the position of a point on the curve changes with time. For a position vector \(\mathbf{r}(t)\), the velocity vector is the derivative with respect to \(t\). This derivative gives insight into the speed and direction of movement along the curve.
For instance, \(\mathbf{r}(t) = \sqrt{t} \mathbf{i} + \sqrt{t} \mathbf{j} - \frac{1}{4}(t+3) \mathbf{k}\) has the velocity vector:\[\mathbf{v}(t) = \frac{d}{dt}\left(\sqrt{t} \mathbf{i} + \sqrt{t} \mathbf{j} - \frac{1}{4}(t+3) \mathbf{k}\right) = \frac{1}{2\sqrt{t}} \mathbf{i} + \frac{1}{2\sqrt{t}} \mathbf{j} - \frac{1}{4} \mathbf{k}\]

• At \(t = 1\), the velocity vector becomes \(\frac{1}{2} \mathbf{i} + \frac{1}{2} \mathbf{j} - \frac{1}{4} \mathbf{k}\).
• This vector indicates how the point moves around on the curve at \(t = 1\).

The comparison between this velocity vector and the gradient vector \((2, 2, -1)\) helps verify whether the curve and the surface are normal to each other at the intersection point. Consistent scalar values between these vectors' components confirm this normalcy.

When a curve and a surface intersect, analysis of their interaction involves checking the angle they make at the intersection point. At the heart of this analysis lies the idea of normalcy—whether the curve encounters the surface perpendicularly. This concept is linked with both the gradient and velocity vectors.

• The intersection in the context of a function \(x^2 + y^2 - z = 3\) and curve \(\mathbf{r}(t)\) occurs at specific points calculated by substituting \(t\) values.
• The point found by setting \(t = 1\) and substituting into \(\mathbf{r}(t)\) gives \((1, 1, -1)\).

At this intersection, the velocity vector of the curve must be checked against the gradient of the surface.
If the velocity vector at this point turns out to be a scalar multiple of the gradient vector, then they are pointing in the same direction, proving that the curve is normal to the surface. Checking component-wise consistency, as shown with scalar \(k\), strengthens this conclusion.