Let $\varphi(x, y)=x+2 \sqrt{y}$ and $C$ be the $V$-shaped curve that starts at $(-1,4)$, makes a line to the origin, then makes a line to end at $(1,4)$

Find the line integral of the gradient of $\varphi$ around the curve $C$.
$$
\int_{C} \nabla \varphi \cdot d s=
$$

Question

Let $\varphi(x, y)=x+2 \sqrt{y}$ and $C$ be the $V$-shaped curve that starts at $(-1,4)$, makes a line to the origin, then makes a line to end at $(1,4)$

Find the line integral of the gradient of $\varphi$ around the curve $C$.
$$
\int_{C} \nabla \varphi \cdot d s=
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Calculate the gradient of the function $φ(x, y)$. ‖ ‖ step 2 ⋮ The gradient of $φ(x, y) = x + 2\sqrt{y}$ is $ abla φ(x, y) = \left( \frac{\partial φ}{\partial x}, \frac{\partial φ}{\partial y} ight) = (1, \frac{1}{\sqrt{y}})$. ‖ ‖ step 3 ⋮ Break the line integral into two parts, one for each segment of the V-shaped curve. ‖ ‖ step 4 ⋮ For the first segment from $(-1, 4)$ to $(0, 0)$, parameterize the curve as $r_1(t) = (-1+t, 4-4t)$, where $0 \leq t \leq 1$. ‖ ‖ step 5 ⋮ Calculate $dr_1/dt = (1, -4)$ and $| abla φ(r_1(t)) \cdot dr_1/dt| = |(1, \frac{1}{\sqrt{4-4t}}) \cdot (1, -4)|$. ‖ ‖ step 6 ⋮ Integrate the dot product over the first segment: $\int_{0}^{1} (1 - \frac{4}{\sqrt{4-4t}}) dt$. ‖ ‖ step 7 ⋮ For the second segment from $(0, 0)$ to $(1, 4)$, parameterize the curve as $r_2(t) = (t, 4t)$, where $0 \leq t \leq 1$. ‖ ‖ step 8 ⋮ Calculate $dr_2/dt = (1, 4)$ and $| abla φ(r_2(t)) \cdot dr_2/dt| = |(1, \frac{1}{\sqrt{4t}}) \cdot (1, 4)|$. ‖ ‖ step 9 ⋮ Integrate the dot product over the second segment: $\int_{0}^{1} (1 + \frac{4}{\sqrt{4t}}) dt$. ‖ ‖ step 10 ⋮ Add the results of the integrals from steps 6 and 9 to find the total line integral around the curve $C$. ‖ ∻Answer∻ ⚹ The line integral of the gradient of $φ$ around the curve $C$ is the sum of the integrals from steps 6 and 9. ⚹ ∻Key Concept∻ ⚹Line Integral of a Gradient⚹ ∻Explanation∻ ⚹The line integral of the gradient of a scalar field along a curve is found by parameterizing the curve, computing the gradient, taking the dot product with the derivative of the parameterization, and integrating over the parameter interval.⚹◊What is the parametric equation for the V-shaped curve C in terms of a parameter t⍭ Generate me a similar question◊

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