## Concept explanation A **vector field** gives every point in space its own **vector**, meaning a direction and a size. You can think of it like an invisible flow map: at one location the field might point mostly right, while at another it might point down or become stronger or weaker. The key idea is that the vector depends on position, so moving to a new point can change both the horizontal and vertical components. ## What you see You’re looking at a grid of arrows that samples the field across the plane. The teal probe marks one chosen location, and the larger teal arrow shows the full vector at that exact point. The blue and green component arrows separate that local vector into horizontal and vertical parts, while the numeric readout lets you compare how those values change as the probe moves. ## Try it yourself - **Drag the probe** to different parts of the grid and watch the large local arrow change direction and length. - **Compare the blue horizontal value and green vertical value** as you move left, right, up, and down. - **Switch the field pattern** to see that different rules create different vector fields across the same space. - **Adjust the grid spacing** to sample the field more coarsely or more densely. - **Change the arrow scale** to make differences in strength easier to notice. - **Reset the probe** and then move it again to test how the vector changes from place to place. ## Concept explanation **Divergence** tells you whether flow in a small neighborhood is, on balance, **spreading out** from a point or **gathering in** toward it. If more motion leaves the little circle than enters it, the divergence is positive. If motion compresses inward, the divergence is negative. When inward and outward tendencies balance, the divergence is zero, even if the field still has some sideways motion. ## What you see You can move the probe circle to sample different parts of the vector field. The colored arrows around the circle show the local tendency near that point: red points inward for negative divergence, gold shows a balanced case, and teal points outward for positive divergence. The tiny particles near the circle make the effect easier to feel visually: they contract inward, hover in place overall, or expand outward depending on the slider. ## Try it yourself - **Drag the divergence slider left** and watch the particles move inward as the neighborhood contracts. - **Place the slider at the middle** and notice that the particles stay roughly at the same radius, showing near-zero divergence. - **Drag the slider right** and see the local arrows flip outward while particles spread away from the center. - **Drag the probe circle** to a new location and compare how the surrounding field looks there versus the local divergence behavior inside the circle. - **Toggle `Show particles`** to switch between the field-and-arrows view and the particle-motion view. - **Press `Reset center`** to return the probe to its starting position and compare the three cases again. ## Concept explanation **Curl** tells you how much a vector field tends to make a tiny object **rotate locally** around a point. A field can push everything in one general direction and still have zero curl if there is no turning effect nearby. What matters is the difference in motion from one side of the wheel to the other: if that pattern twists the wheel clockwise or counterclockwise, the curl is nonzero. ## What you see The arrows show the field around the draggable wheel center. The slider changes the local rotational strength from negative values (clockwise tendency) through `0` to positive values (counterclockwise tendency). The pinwheel spins to match that local turning effect, and the optional drift mode adds overall motion so you can compare “moving along” with “being made to turn.” ## Try it yourself - **Drag the wheel** to different parts of the field and watch how the nearby arrows still organize around the center. - **Move the rotational strength slider** toward negative values and notice the wheel reverse into clockwise spin. - **Set the slider to `0`** and check that the wheel stops, showing zero local turning. - **Increase the slider to positive values** and see the counterclockwise rotation speed up. - **Switch to `Curl + drift`** and compare the background flow with the wheel’s spin so you can separate overall motion from local rotation. - **Press `Reset wheel`** to return the probe to the middle and test the effect again from a clean starting point. ## Concept explanation **Flux** measures how much of a **vector field** passes through a chosen boundary. For a line segment in 2D, the important comparison is between the field direction and the segment’s **normal vector**: when the field points strongly across the segment, flux is large; when it runs almost parallel to the segment, the crossing is small and the flux drops toward `0`. Reversing the normal also reverses the sign, because you are measuring flow through the boundary in the opposite direction. ## What you see You’re looking at a vector field drawn with blue arrows, plus a movable white line segment with endpoints `A` and `B`. The teal arrow shows the current normal direction `n̂`. Near the segment, arrows are recolored to show how they contribute to flux: green for positive crossing, red for negative crossing, and gold for nearly tangential flow. The single flux meter in the corner tracks the net signed crossing through the segment. ## Try it yourself - **Drag endpoint `A` or `B`** to rotate the segment and notice how the flux meter grows when the segment faces across the flow. - **Line the segment up with the field arrows** and watch the gold near-tangential arrows appear while the flux meter moves toward `0`. - **Use the field strength slider** to scale every vector and see the flux magnitude increase as the flow becomes stronger. - **Switch the field pattern** to compare a uniform flow, a shear flow, and a swirl, then look for places where the same segment orientation gives different flux. - **Press the Flip normal button** to reverse `n̂` and see the meter change sign even though the geometry stays the same. ## Concept explanation In a 2D vector field, **divergence** measures whether the flow near a point is locally spreading outward or squeezing inward, while **curl** measures whether the flow is locally trying to spin around that point. These are different properties: a field can spread without rotating, rotate without spreading, do both at once, or do neither. By controlling them separately here, you can see that local expansion/contraction and local rotation are independent behaviors. ## What you see You’re looking at a vector field sampled around a movable probe. The arrows show the local flow pattern around that probe point. Beside it, the pulsing ring represents local expansion or contraction, and the spinning wheel represents local rotation. Because both indicators respond to different sliders, you can compare what changes in the field shape with what changes in each local diagnostic. ## Try it yourself - **Set both sliders to `0`** and notice that the arrows around the probe lose both spreading and spinning. - **Increase only the outward / inward slider** to create pure divergence, and watch the ring pulse while the wheel stays still. - **Increase only the rotational strength slider** to create pure curl, and watch the wheel spin while the ring no longer signals expansion or contraction. - **Make both sliders nonzero** to combine spreading and spinning in the same local neighborhood. - **Use negative values** on either slider to reverse the behavior: inward contraction for divergence, or opposite spin direction for curl. - **Drag the probe** to different places in the field and see that the same two local properties are being measured at the new location. ## Concept explanation A **net flux** measures the overall field crossing a **closed boundary**: outward crossings count positive, inward crossings count negative. If the total is positive, more field leaves the region than enters; if it is negative, more enters than leaves. This connects to **divergence**, because net flux over the loop reflects the accumulated tendency of the field inside the enclosed area to act like a source, a sink, or neither. ## What you see You’re looking at a 2D vector field with a movable circular loop. Short animated markers along the loop show the local crossing direction: red markers indicate flow leaving the loop, while gold markers indicate flow entering it. The gauge on the right sums all those boundary crossings into one signed total, so you can compare what happens when the loop surrounds outward-spreading behavior, inward-pulling behavior, or a nearly balanced field. ## Try it yourself - **Drag the loop** around the field and watch how the boundary markers change as different parts of the field fall inside. - **Move the Field pattern slider toward `1`** to create an outward-source pattern and notice the gauge become positive when the loop encloses that source. - **Move the Field pattern slider toward `-1`** to create an inward-sink pattern and see the gauge turn negative as more field enters than leaves. - **Set the Field pattern slider near `0`** and observe how the entering and leaving markers nearly balance, making the net flux stay close to zero. - **Adjust the Loop size slider** to compare a small enclosed area with a larger one and see how the total flux reflects how much divergence is captured inside. - **Use Reset loop** to return to the starting setup and test the same idea again from a clean position.