📝 Gradient perpendicular to level curves (13 MCQs)

📖 Explanation: The gradient vector $\nabla f$ is always orthogonal to level curves and points in the direction of steepest ascent. Walking tangent yields zero elevation change, while walking opposite descends. Therefore, ascending most steeply requires moving parallel to the gradient, confirming the fundamental geometric relationship between gradients and level sets.

📖 Explanation: At critical points where $\nabla f = \mathbf{0}$, the implicit function theorem fails and the level set may have singularities like crossings. Here $\nabla f = \langle 2x, -2y \rangle = \langle 0,0 \rangle$ at the origin, making the normal vector undefined. This highlights that gradient-normality assumes non-vanishing gradients and smooth manifolds.

📖 Explanation: A zero directional derivative implies $\nabla T \cdot \mathbf{u} = 0$, meaning the gradient is orthogonal to $\mathbf{u}$. Since $\mathbf{u}$ points along $\langle 1,1 \rangle$, any vector perpendicular to it must be scalar multiples of $\langle 1,-1 \rangle$. This uses the geometric definition directly, bypassing computation and reinforcing orthogonality concepts over rote calculation.

📖 Explanation: The error confuses orthogonal directions with descent directions. Movement perpendicular to $\nabla f$ is tangent to the level curve, yielding zero instantaneous rate of change. Descent requires moving opposite to $\nabla f$, not perpendicular to it. This misconception arises from misapplying orthogonality and failing to distinguish between tangential and radial behaviors near level sets.

📖 Explanation: Graphically, gradient direction is normal to level curves, and magnitude correlates with curve density. Tighter spacing indicates larger $\|\nabla f\|$. Thus, a correct gradient arrow must be both perpendicular to the local level curve and oriented toward regions of higher density. Tangent arrows represent zero change, while bisectors lack geometric justification in this context.

📖 Explanation: Vertical translation does not alter partial derivatives because constants vanish upon differentiation. Thus $\nabla g = \nabla f$, and level curves of $g=k$ correspond exactly to $f=k-c$, preserving geometry. The gradient’s normality depends solely on local slope structure, not absolute height. This reinforces that gradients encode rate-of-change information invariant under additive constants.

📖 Explanation: The Lagrange condition $\nabla f = \lambda \nabla g$ is necessary for smooth constrained extrema when $\nabla g \neq \mathbf{0}$. Non-parallel nonzero gradients mean the objective’s steepest ascent isn’t aligned with the constraint’s normal, allowing feasible movement that improves $f$. Hence, such a point cannot be a local max/min under regularity conditions, highlighting the geometric necessity of gradient alignment.

📖 Explanation: Wind flowing from high to low pressure moves opposite to $\nabla P$, since gradient points toward greatest increase. Northward wind implies decreasing pressure northward, so pressure increases southward. Thus $\nabla P$ points south. This applies the gradient-normal-to-level-curves principle in a real-world meteorological context, linking abstract math to physical vector fields and directional interpretation.

📖 Explanation: Even small gradient errors break exact orthogonality. The true normal intersects level curves perpendicularly, but an inaccurate gradient defines a skewed line that crosses curves obliquely. In iterative methods like gradient descent or contour tracking, this accumulates error. This question tests sensitivity analysis and understanding that geometric properties like normality are not robust to numerical imprecision.

📖 Explanation: By chain rule, \nabla h = \phi'(f) \nabla f. Since $\phi$ is strictly increasing, \phi' > 0, preserving direction. Magnitude scales by \phi', reflecting how reparameterization stretches or compresses the function vertically without altering level set geometry. This shows gradient direction encodes level curve orientation independently of monotonic transformations, while magnitude depends on scaling.

📖 Explanation: Maximal increase occurs along $\nabla f$; zero change occurs along directions tangent to the level curve. By definition, gradient is normal to level curves, so these directions must be orthogonal. This pure conceptual question strips away coordinates to test foundational understanding of gradient geometry as intrinsic to scalar fields, independent of representation.

📖 Explanation: For $f=r$, $\nabla f = \langle x/r, y/r \rangle$, which has unit magnitude. Though circles get farther apart radially, the function increases linearly with radius, so slope remains constant. Steepness relates to $df/dr$, not curvature of level sets. This challenges the misconception that level curve spacing alone determines gradient magnitude without considering functional form.

📖 Explanation: Zero change along $\gamma$ implies $\nabla V \perp \gamma$. Since $\delta \perp \gamma$, $\nabla V$ must be parallel to $\delta$. The nonzero change along $\delta$ confirms $\nabla V \neq \mathbf{0}$. This experimental scenario applies gradient-normality inversely: observed null directional derivatives reveal gradient orientation. It integrates measurement interpretation with vector calculus principles in a physics context.