The Meaning of Gradient, 𝛁 Propulsor System Laboratory Department of Naval Architecture & Ocean Engineering Pusan National University Yoo, Gwang Yeol

Contents 01. Gradient? 02. Gradient and graphs 03. Gradient and contour maps

𝒇(𝒙,𝒚,𝒛) 의 변화율 = 𝜵𝒇= 𝝏𝒇 𝝏𝒙 𝒊 , 𝝏𝒇 𝝏𝒚 𝒋 , 𝝏𝒇 𝝏𝒛 𝒌 𝑻 Gradient? 구배? 기울기? 변화율 경사도? 각 성분의 편미분 열벡터 𝒇(𝒙,𝒚,𝒛) 의 변화율 = 𝜵𝒇= 𝝏𝒇 𝝏𝒙 𝒊 , 𝝏𝒇 𝝏𝒚 𝒋 , 𝝏𝒇 𝝏𝒛 𝒌 𝑻 델(del)연산자

Gradient and graphs 𝒇 𝒙,𝒚 = 𝒙 𝟐 + 𝒚 𝟐 → 𝒛= 𝒙 𝟐 + 𝒚 𝟐 Considering, 𝒇 𝒙,𝒚 = 𝒙 𝟐 + 𝒚 𝟐 𝛻𝑓 𝑥,𝑦 = 𝜕𝑓 𝜕𝑥 𝜕𝑓 𝜕𝑦 = 𝜕 𝑥 2 + 𝑦 2 𝜕𝑥 𝜕 𝑥 2 + 𝑦 2 𝜕𝑦 = 2𝑥 2𝑦 𝒇 𝒙,𝒚 = 𝒙 𝟐 + 𝒚 𝟐 → 𝒛= 𝒙 𝟐 + 𝒚 𝟐 o

Gradient means direction of steepest ascent. Gradient and graphs 𝒛= 𝒙 𝟐 + 𝒚 𝟐 𝜵𝒇 𝒙,𝒚 = 𝟐𝒙 𝟐𝒚 Gradient means direction of steepest ascent.

Gradient and contour maps Considering, 𝒇 𝒙,𝒚 = 𝒙 𝟐 + 𝒚 𝟐 y 𝑥 2 + 𝑦 2 =6 𝑥 2 + 𝑦 2 =4 𝑥 2 + 𝑦 2 =2 x Gradient is always perpendicular to contour lines.

Gradient and contour maps Considering, 𝒇 𝒙,𝒚 =𝒙𝒚

Thank you for your attention! Any questions?