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A gradient vector is like a compass for multivariable functions. It points in the steepest uphill direction of a function at a given point. This means that when you're at a hill, the gradient vector tells you which way is straight uphill. For the function \( h(x, y) = e^x \sin y \), the gradient vector at point \( P(1, \frac{\pi}{2}) \) is found using the partial derivatives calculated at this point:

• The partial derivative with respect to \( x \) gives the gradient's \( x \)-component: \( e \).
• The partial derivative with respect to \( y \) gives the gradient's \( y \)-component: \( 0 \).

So, the gradient vector at \( P(1, \frac{\pi}{2}) \) is \( (e, 0) \). This vector shows that the function is changing most rapidly in the positive \( x \)-direction, as indicated by the non-zero value \( e \).

Partial derivatives are like the slices of a multivariable function, showing how the function changes with respect to one variable at a time while keeping the others constant. In the exercise, we look at two partial derivatives:

• With respect to \( x \), we have \( \frac{\partial h}{\partial x} = e^x \sin y \). This tells us how \( h \) changes as \( x \) changes, while \( y \) stays fixed.
• With respect to \( y \), \( \frac{\partial h}{\partial y} = e^x \cos y \). Similarly, this shows how \( h \) varies as \( y \) changes, holding \( x \) constant.

By evaluating these derivatives at a specific point \( P(1, \frac{\pi}{2}) \), we determine how the function behaves locally around that point. Thus, partial derivatives play a crucial role in constructing the gradient vector, which captures the overall directional change.

A unit vector is a vector with a magnitude of 1. It serves as a direction indicator without affecting the scale or length of what it's applied to. When considering the direction for a directional derivative, the unit vector ensures that only the direction contributes, not any scaling.In the exercise, the directional vector \( \mathbf{v} = -\mathbf{i} = (-1, 0) \) is already a unit vector. The magnitude is calculated as:\[\| \mathbf{v} \| = \sqrt{(-1)^2 + 0^2} = 1.\]This means that \( \mathbf{v} \) requires no further adjustment or scaling. Because \( \mathbf{v} \) maintains a unit length, it allows us to compute the directional derivative accurately by multiplying the gradient vector \( abla h \) with \( \mathbf{v} \) through a dot product, focusing solely on direction.